LMFDB - Embedded newform 336.6.h.b.239.5

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			239.5
Character	$\chi$	$=$	336.239
Dual form			336.6.h.b.239.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.6996 - 7.43773i) q^{3} -101.234i q^{5} -49.0000i q^{7} +(132.360 + 203.789i) q^{9} +O(q^{10})$	\(q+(-13.6996 - 7.43773i) q^{3} -101.234i q^{5} -49.0000i q^{7} +(132.360 + 203.789i) q^{9} +563.943 q^{11} -261.477 q^{13} +(-752.954 + 1386.87i) q^{15} +2058.95i q^{17} +978.317i q^{19} +(-364.449 + 671.282i) q^{21} +2208.61 q^{23} -7123.40 q^{25} +(-297.564 - 3776.29i) q^{27} +5604.45i q^{29} +4737.20i q^{31} +(-7725.81 - 4194.46i) q^{33} -4960.48 q^{35} +10817.2 q^{37} +(3582.14 + 1944.80i) q^{39} +17703.6i q^{41} -12329.1i q^{43} +(20630.4 - 13399.4i) q^{45} -6701.70 q^{47} -2401.00 q^{49} +(15313.9 - 28206.8i) q^{51} +24845.8i q^{53} -57090.4i q^{55} +(7276.46 - 13402.6i) q^{57} -21403.3 q^{59} +4697.43 q^{61} +(9985.64 - 6485.65i) q^{63} +26470.5i q^{65} +49298.4i q^{67} +(-30257.2 - 16427.1i) q^{69} +42579.5 q^{71} -1240.19 q^{73} +(97588.0 + 52981.9i) q^{75} -27633.2i q^{77} +63235.3i q^{79} +(-24010.5 + 53947.0i) q^{81} +62227.9 q^{83} +208436. q^{85} +(41684.4 - 76779.0i) q^{87} +27544.1i q^{89} +12812.4i q^{91} +(35234.0 - 64897.9i) q^{93} +99039.3 q^{95} -178615. q^{97} +(74643.6 + 114925. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−13.6996	−	7.43773i	−0.878832	−	0.477131i
$3$	−13.6996	−	7.43773i	−0.878832	−	0.477131i
$4$		0			0
$5$		−	101.234i		−	1.81094i	−0.424415	−	0.905468i	$-0.639520\pi$
$5$		−	101.234i		−	1.81094i	0.424415	−	0.905468i	$-0.360480\pi$
$6$		0			0
$7$		−	49.0000i		−	0.377964i
$7$		−	49.0000i		−	0.377964i
$8$		0			0
$9$	132.360	+	203.789i	0.544692	+	0.838636i
$10$		0			0
$11$	563.943			1.40525			0.702625	−	0.711561i	$-0.252011\pi$
$11$	563.943			1.40525			0.702625	+	0.711561i	$0.252011\pi$
$12$		0			0
$13$	−261.477			−0.429117			−0.214558	−	0.976711i	$-0.568831\pi$
$13$	−261.477			−0.429117			−0.214558	+	0.976711i	$0.568831\pi$
$14$		0			0
$15$	−752.954	+	1386.87i	−0.864053	+	1.59151i
$16$		0			0
$17$			2058.95i			1.72792i	0.503563	+	0.863959i	$0.332022\pi$
$17$			2058.95i			1.72792i	−0.503563	+	0.863959i	$0.667978\pi$
$18$		0			0
$19$			978.317i			0.621721i	0.950456	+	0.310861i	$0.100617\pi$
$19$			978.317i			0.621721i	−0.950456	+	0.310861i	$0.899383\pi$
$20$		0			0
$21$	−364.449	+	671.282i	−0.180338	+	0.332167i
$22$		0			0
$23$	2208.61			0.870563			0.435282	−	0.900294i	$-0.356649\pi$
$23$	2208.61			0.870563			0.435282	+	0.900294i	$0.356649\pi$
$24$		0			0
$25$	−7123.40			−2.27949
$26$		0			0
$27$	−297.564	−	3776.29i	−0.0785544	−	0.996910i
$28$		0			0
$29$			5604.45i			1.23748i	0.785596	+	0.618740i	$0.212356\pi$
$29$			5604.45i			1.23748i	−0.785596	+	0.618740i	$0.787644\pi$
$30$		0			0
$31$			4737.20i			0.885355i	0.896681	+	0.442677i	$0.145971\pi$
$31$			4737.20i			0.885355i	−0.896681	+	0.442677i	$0.854029\pi$
$32$		0			0
$33$	−7725.81	−	4194.46i	−1.23498	−	0.670488i
$34$		0			0
$35$	−4960.48			−0.684469
$36$		0			0
$37$	10817.2			1.29900			0.649502	−	0.760360i	$-0.274978\pi$
$37$	10817.2			1.29900			0.649502	+	0.760360i	$0.274978\pi$
$38$		0			0
$39$	3582.14	+	1944.80i	0.377122	+	0.204745i
$40$		0			0
$41$			17703.6i			1.64476i	0.568940	+	0.822379i	$0.307354\pi$
$41$			17703.6i			1.64476i	−0.568940	+	0.822379i	$0.692646\pi$
$42$		0			0
$43$		−	12329.1i		−	1.01686i	−0.861104	−	0.508429i	$-0.830226\pi$
$43$		−	12329.1i		−	1.01686i	0.861104	−	0.508429i	$-0.169774\pi$
$44$		0			0
$45$	20630.4	−	13399.4i	1.51872	−	0.986403i
$46$		0			0
$47$	−6701.70			−0.442528			−0.221264	−	0.975214i	$-0.571018\pi$
$47$	−6701.70			−0.442528			−0.221264	+	0.975214i	$0.571018\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	15313.9	−	28206.8i	0.824442	−	1.51855i
$52$		0			0
$53$			24845.8i			1.21496i	0.794334	+	0.607482i	$0.207820\pi$
$53$			24845.8i			1.21496i	−0.794334	+	0.607482i	$0.792180\pi$
$54$		0			0
$55$		−	57090.4i		−	2.54482i
$56$		0			0
$57$	7276.46	−	13402.6i	0.296642	−	0.546389i
$58$		0			0
$59$	−21403.3			−0.800480			−0.400240	−	0.916410i	$-0.631073\pi$
$59$	−21403.3			−0.800480			−0.400240	+	0.916410i	$0.631073\pi$
$60$		0			0
$61$	4697.43			0.161635			0.0808176	−	0.996729i	$-0.474247\pi$
$61$	4697.43			0.161635			0.0808176	+	0.996729i	$0.474247\pi$
$62$		0			0
$63$	9985.64	−	6485.65i	0.316975	−	0.205874i
$64$		0			0
$65$			26470.5i			0.777103i
$66$		0			0
$67$			49298.4i			1.34167i	0.741606	+	0.670835i	$0.234064\pi$
$67$			49298.4i			1.34167i	−0.741606	+	0.670835i	$0.765936\pi$
$68$		0			0
$69$	−30257.2	−	16427.1i	−0.765079	−	0.415373i
$70$		0			0
$71$	42579.5			1.00243			0.501216	−	0.865322i	$-0.332886\pi$
$71$	42579.5			1.00243			0.501216	+	0.865322i	$0.332886\pi$
$72$		0			0
$73$	−1240.19			−0.0272384			−0.0136192	−	0.999907i	$-0.504335\pi$
$73$	−1240.19			−0.0272384			−0.0136192	+	0.999907i	$0.504335\pi$
$74$		0			0
$75$	97588.0	+	52981.9i	2.00329	+	1.08761i
$76$		0			0
$77$		−	27633.2i		−	0.531134i
$78$		0			0
$79$			63235.3i			1.13997i	0.821657	+	0.569983i	$0.193050\pi$
$79$			63235.3i			1.13997i	−0.821657	+	0.569983i	$0.806950\pi$
$80$		0			0
$81$	−24010.5	+	53947.0i	−0.406620	+	0.913597i
$82$		0			0
$83$	62227.9			0.991493			0.495746	−	0.868467i	$-0.334895\pi$
$83$	62227.9			0.991493			0.495746	+	0.868467i	$0.334895\pi$
$84$		0			0
$85$	208436.			3.12915
$86$		0			0
$87$	41684.4	−	76779.0i	0.590440	−	1.08754i
$88$		0			0
$89$			27544.1i			0.368598i	0.982870	+	0.184299i	$0.0590015\pi$
$89$			27544.1i			0.368598i	−0.982870	+	0.184299i	$0.940998\pi$
$90$		0			0
$91$			12812.4i			0.162191i
$92$		0			0
$93$	35234.0	−	64897.9i	0.422430	−	0.778078i
$94$		0			0
$95$	99039.3			1.12590
$96$		0			0
$97$	−178615.			−1.92748			−0.963739	−	0.266847i	$-0.914018\pi$
$97$	−178615.			−1.92748			−0.963739	+	0.266847i	$0.914018\pi$
$98$		0			0
$99$	74643.6	+	114925.i	0.765429	+	1.17849i
$100$		0			0
$101$			56370.2i			0.549853i	0.961465	+	0.274926i	$0.0886535\pi$
$101$			56370.2i			0.549853i	−0.961465	+	0.274926i	$0.911347\pi$
$102$		0			0
$103$		−	161998.i		−	1.50458i	−0.658830	−	0.752291i	$-0.728949\pi$
$103$		−	161998.i		−	1.50458i	0.658830	−	0.752291i	$-0.271051\pi$
$104$		0			0
$105$	67956.8	+	36894.8i	0.601534	+	0.326581i
$106$		0			0
$107$	198843.			1.67900			0.839499	−	0.543362i	$-0.182849\pi$
$107$	198843.			1.67900			0.839499	+	0.543362i	$0.182849\pi$
$108$		0			0
$109$	−25080.3			−0.202194			−0.101097	−	0.994877i	$-0.532235\pi$
$109$	−25080.3			−0.202194			−0.101097	+	0.994877i	$0.532235\pi$
$110$		0			0
$111$	−148192.	−	80455.4i	−1.14161	−	0.619794i
$112$		0			0
$113$		−	3774.58i		−	0.0278082i	−0.999903	−	0.0139041i	$-0.995574\pi$
$113$		−	3774.58i		−	0.0278082i	0.999903	−	0.0139041i	$-0.00442596\pi$
$114$		0			0
$115$		−	223588.i		−	1.57653i
$116$		0			0
$117$	−34609.2	−	53286.0i	−0.233737	−	0.359873i
$118$		0			0
$119$	100888.			0.653091
$120$		0			0
$121$	156980.			0.974725
$122$		0			0
$123$	131675.	−	242533.i	0.784765	−	1.44547i
$124$		0			0
$125$			404775.i			2.31707i
$126$		0			0
$127$			180250.i			0.991666i	0.868418	+	0.495833i	$0.165137\pi$
$127$			180250.i			0.991666i	−0.868418	+	0.495833i	$0.834863\pi$
$128$		0			0
$129$	−91700.5	+	168904.i	−0.485174	+	0.893648i
$130$		0			0
$131$	−80733.0			−0.411029			−0.205515	−	0.978654i	$-0.565887\pi$
$131$	−80733.0			−0.411029			−0.205515	+	0.978654i	$0.565887\pi$
$132$		0			0
$133$	47937.5			0.234988
$134$		0			0
$135$	−382290.	+	30123.7i	−1.80534	+	0.142257i
$136$		0			0
$137$		−	162782.i		−	0.740979i	−0.928837	−	0.370490i	$-0.879190\pi$
$137$		−	162782.i		−	0.740979i	0.928837	−	0.370490i	$-0.120810\pi$
$138$		0			0
$139$		−	90111.8i		−	0.395589i	−0.980243	−	0.197795i	$-0.936622\pi$
$139$		−	90111.8i		−	0.395589i	0.980243	−	0.197795i	$-0.0633779\pi$
$140$		0			0
$141$	91810.9	+	49845.5i	0.388908	+	0.211144i
$142$		0			0
$143$	−147458.			−0.603016
$144$		0			0
$145$	567363.			2.24100
$146$		0			0
$147$	32892.8	+	17858.0i	0.125547	+	0.0681615i
$148$		0			0
$149$		−	326696.i		−	1.20553i	−0.797919	−	0.602765i	$-0.794066\pi$
$149$		−	326696.i		−	1.20553i	0.797919	−	0.602765i	$-0.205934\pi$
$150$		0			0
$151$		−	317385.i		−	1.13278i	−0.824139	−	0.566388i	$-0.808340\pi$
$151$		−	317385.i		−	1.13278i	0.824139	−	0.566388i	$-0.191660\pi$
$152$		0			0
$153$	−419590.	+	272523.i	−1.44909	+	0.941183i
$154$		0			0
$155$	479567.			1.60332
$156$		0			0
$157$	79242.2			0.256571			0.128285	−	0.991737i	$-0.459053\pi$
$157$	79242.2			0.256571			0.128285	+	0.991737i	$0.459053\pi$
$158$		0			0
$159$	184796.	−	340378.i	0.579697	−	1.06775i
$160$		0			0
$161$		−	108222.i		−	0.329042i
$162$		0			0
$163$			60963.4i			0.179722i	0.995954	+	0.0898608i	$0.0286422\pi$
$163$			60963.4i			0.179722i	−0.995954	+	0.0898608i	$0.971358\pi$
$164$		0			0
$165$	−424623.	+	782118.i	−1.21421	+	2.23647i
$166$		0			0
$167$	248987.			0.690853			0.345427	−	0.938446i	$-0.387734\pi$
$167$	248987.			0.690853			0.345427	+	0.938446i	$0.387734\pi$
$168$		0			0
$169$	−302923.			−0.815859
$170$		0			0
$171$	−199370.	+	129490.i	−0.521398	+	0.338647i
$172$		0			0
$173$		−	10631.2i		−	0.0270064i	−0.999909	−	0.0135032i	$-0.995702\pi$
$173$		−	10631.2i		−	0.0270064i	0.999909	−	0.0135032i	$-0.00429834\pi$
$174$		0			0
$175$			349046.i			0.861565i
$176$		0			0
$177$	293217.	+	159192.i	0.703488	+	0.381934i
$178$		0			0
$179$	−549045.			−1.28078			−0.640391	−	0.768049i	$-0.721228\pi$
$179$	−549045.			−1.28078			−0.640391	+	0.768049i	$0.721228\pi$
$180$		0			0
$181$	−784501.			−1.77991			−0.889953	−	0.456052i	$-0.849263\pi$
$181$	−784501.			−1.77991			−0.889953	+	0.456052i	$0.849263\pi$
$182$		0			0
$183$	−64353.2	−	34938.3i	−0.142050	−	0.0771212i
$184$		0			0
$185$		−	1.09507e6i		−	2.35241i
$186$		0			0
$187$			1.16113e6i			2.42815i
$188$		0			0
$189$	−185038.	+	14580.6i	−0.376796	+	0.0296908i
$190$		0			0
$191$	158195.			0.313768			0.156884	−	0.987617i	$-0.449855\pi$
$191$	158195.			0.313768			0.156884	+	0.987617i	$0.449855\pi$
$192$		0			0
$193$	−200874.			−0.388178			−0.194089	−	0.980984i	$-0.562175\pi$
$193$	−200874.			−0.388178			−0.194089	+	0.980984i	$0.562175\pi$
$194$		0			0
$195$	196880.	−	362636.i	0.370780	−	0.682943i
$196$		0			0
$197$			669487.i			1.22907i	0.788890	+	0.614535i	$0.210656\pi$
$197$			669487.i			1.22907i	−0.788890	+	0.614535i	$0.789344\pi$
$198$		0			0
$199$		−	132738.i		−	0.237608i	−0.992918	−	0.118804i	$-0.962094\pi$
$199$		−	132738.i		−	0.237608i	0.992918	−	0.118804i	$-0.0379061\pi$
$200$		0			0
$201$	366668.	−	675370.i	0.640152	−	1.17910i
$202$		0			0
$203$	274618.			0.467723
$204$		0			0
$205$	1.79221e6			2.97855
$206$		0			0
$207$	292333.	+	450090.i	0.474189	+	0.730086i
$208$		0			0
$209$			551715.i			0.873673i
$210$		0			0
$211$			305295.i			0.472077i	0.971744	+	0.236039i	$0.0758492\pi$
$211$			305295.i			0.472077i	−0.971744	+	0.236039i	$0.924151\pi$
$212$		0			0
$213$	−583324.	−	316695.i	−0.880970	−	0.478291i
$214$		0			0
$215$	−1.24813e6			−1.84146
$216$		0			0
$217$	232123.			0.334633
$218$		0			0
$219$	16990.2	+	9224.21i	0.0239380	+	0.0129963i
$220$		0			0
$221$		−	538368.i		−	0.741478i
$222$		0			0
$223$			780141.i			1.05054i	0.850937	+	0.525268i	$0.176035\pi$
$223$			780141.i			1.05054i	−0.850937	+	0.525268i	$0.823965\pi$
$224$		0			0
$225$	−942855.	−	1.45167e6i	−1.24162	−	1.91166i
$226$		0			0
$227$	−686314.			−0.884011			−0.442006	−	0.897012i	$-0.645733\pi$
$227$	−686314.			−0.884011			−0.442006	+	0.897012i	$0.645733\pi$
$228$		0			0
$229$	−620027.			−0.781307			−0.390654	−	0.920538i	$-0.627751\pi$
$229$	−620027.			−0.781307			−0.390654	+	0.920538i	$0.627751\pi$
$230$		0			0
$231$	−205528.	+	378565.i	−0.253421	+	0.466778i
$232$		0			0
$233$		−	449666.i		−	0.542625i	−0.962491	−	0.271313i	$-0.912542\pi$
$233$		−	449666.i		−	0.542625i	0.962491	−	0.271313i	$-0.0874577\pi$
$234$		0			0
$235$			678442.i			0.801389i
$236$		0			0
$237$	470327.	−	866301.i	0.543913	−	1.00184i
$238$		0			0
$239$	298507.			0.338034			0.169017	−	0.985613i	$-0.445941\pi$
$239$	298507.			0.338034			0.169017	+	0.985613i	$0.445941\pi$
$240$		0			0
$241$	577041.			0.639976			0.319988	−	0.947422i	$-0.396321\pi$
$241$	577041.			0.639976			0.319988	+	0.947422i	$0.396321\pi$
$242$		0			0
$243$	730179.	−	560471.i	0.793256	−	0.608888i
$244$		0			0
$245$			243064.i			0.258705i
$246$		0			0
$247$		−	255808.i		−	0.266791i
$248$		0			0
$249$	−852499.	−	462834.i	−0.871356	−	0.473072i
$250$		0			0
$251$	−1.08471e6			−1.08675			−0.543375	−	0.839490i	$-0.682854\pi$
$251$	−1.08471e6			−1.08675			−0.543375	+	0.839490i	$0.682854\pi$
$252$		0			0
$253$	1.24553e6			1.22336
$254$		0			0
$255$	−2.85550e6	−	1.55029e6i	−2.74999	−	1.49301i
$256$		0			0
$257$		−	735754.i		−	0.694864i	−0.937705	−	0.347432i	$-0.887054\pi$
$257$		−	735754.i		−	0.694864i	0.937705	−	0.347432i	$-0.112946\pi$
$258$		0			0
$259$		−	530042.i		−	0.490977i
$260$		0			0
$261$	−1.14212e6	+	741807.i	−1.03779	+	0.674046i
$262$		0			0
$263$	728009.			0.649004			0.324502	−	0.945885i	$-0.394803\pi$
$263$	728009.			0.649004			0.324502	+	0.945885i	$0.394803\pi$
$264$		0			0
$265$	2.51525e6			2.20022
$266$		0			0
$267$	204866.	−	377344.i	0.175870	−	0.323936i
$268$		0			0
$269$			371877.i			0.313342i	0.987651	+	0.156671i	$0.0500763\pi$
$269$			371877.i			0.313342i	−0.987651	+	0.156671i	$0.949924\pi$
$270$		0			0
$271$		−	403357.i		−	0.333631i	−0.985988	−	0.166816i	$-0.946652\pi$
$271$		−	403357.i		−	0.333631i	0.985988	−	0.166816i	$-0.0533485\pi$
$272$		0			0
$273$	95295.1	−	175525.i	0.0773863	−	0.142539i
$274$		0			0
$275$	−4.01719e6			−3.20325
$276$		0			0
$277$	274707.			0.215115			0.107557	−	0.994199i	$-0.465697\pi$
$277$	274707.			0.215115			0.107557	+	0.994199i	$0.465697\pi$
$278$		0			0
$279$	−965387.	+	627017.i	−0.742490	+	0.482246i
$280$		0			0
$281$			1.26819e6i			0.958115i	0.877784	+	0.479057i	$0.159021\pi$
$281$			1.26819e6i			0.958115i	−0.877784	+	0.479057i	$0.840979\pi$
$282$		0			0
$283$		−	827068.i		−	0.613868i	−0.951731	−	0.306934i	$-0.900697\pi$
$283$		−	827068.i		−	0.613868i	0.951731	−	0.306934i	$-0.0993031\pi$
$284$		0			0
$285$	−1.35680e6	−	736628.i	−0.989474	−	0.537200i
$286$		0			0
$287$	867476.			0.621660
$288$		0			0
$289$	−2.81941e6			−1.98570
$290$		0			0
$291$	2.44697e6	+	1.32849e6i	1.69393	+	0.919659i
$292$		0			0
$293$			523372.i			0.356157i	0.984016	+	0.178079i	$0.0569881\pi$
$293$			523372.i			0.356157i	−0.984016	+	0.178079i	$0.943012\pi$
$294$		0			0
$295$			2.16675e6i			1.44962i
$296$		0			0
$297$	−167809.	−	2.12961e6i	−0.110388	−	1.40091i
$298$		0			0
$299$	−577502.			−0.373573
$300$		0			0
$301$	−604126.			−0.384336
$302$		0			0
$303$	419267.	−	772252.i	0.262352	−	0.483228i
$304$		0			0
$305$		−	475542.i		−	0.292711i
$306$		0			0
$307$			1.55949e6i			0.944358i	0.881503	+	0.472179i	$0.156532\pi$
$307$			1.55949e6i			0.944358i	−0.881503	+	0.472179i	$0.843468\pi$
$308$		0			0
$309$	−1.20490e6	+	2.21931e6i	−0.717883	+	1.32228i
$310$		0			0
$311$	1.05288e6			0.617273			0.308637	−	0.951180i	$-0.400127\pi$
$311$	1.05288e6			0.617273			0.308637	+	0.951180i	$0.400127\pi$
$312$		0			0
$313$	−621741.			−0.358714			−0.179357	−	0.983784i	$-0.557402\pi$
$313$	−621741.			−0.358714			−0.179357	+	0.983784i	$0.557402\pi$
$314$		0			0
$315$	−656571.	−	1.01089e6i	−0.372825	−	0.574020i
$316$		0			0
$317$			564641.i			0.315591i	0.987472	+	0.157795i	$0.0504386\pi$
$317$			564641.i			0.315591i	−0.987472	+	0.157795i	$0.949561\pi$
$318$		0			0
$319$			3.16059e6i			1.73897i
$320$		0			0
$321$	−2.72407e6	−	1.47894e6i	−1.47556	−	0.801101i
$322$		0			0
$323$	−2.01430e6			−1.07428
$324$		0			0
$325$	1.86261e6			0.978166
$326$		0			0
$327$	343592.	+	186541.i	0.177694	+	0.0964728i
$328$		0			0
$329$			328383.i			0.167260i
$330$		0			0
$331$			2.64530e6i			1.32710i	0.748130	+	0.663552i	$0.230952\pi$
$331$			2.64530e6i			1.32710i	−0.748130	+	0.663552i	$0.769048\pi$
$332$		0			0
$333$	1.43177e6	+	2.20442e6i	0.707557	+	1.08939i
$334$		0			0
$335$	4.99069e6			2.42968
$336$		0			0
$337$	−1.18056e6			−0.566254			−0.283127	−	0.959082i	$-0.591372\pi$
$337$	−1.18056e6			−0.566254			−0.283127	+	0.959082i	$0.591372\pi$
$338$		0			0
$339$	−28074.3	+	51710.4i	−0.0132682	+	0.0244388i
$340$		0			0
$341$			2.67151e6i			1.24414i
$342$		0			0
$343$			117649.i			0.0539949i
$344$		0			0
$345$	−1.66299e6	+	3.06307e6i	−0.752213	+	1.38551i
$346$		0			0
$347$	−2.60432e6			−1.16110			−0.580551	−	0.814224i	$-0.697163\pi$
$347$	−2.60432e6			−1.16110			−0.580551	+	0.814224i	$0.697163\pi$
$348$		0			0
$349$	4.04756e6			1.77881			0.889405	−	0.457120i	$-0.151119\pi$
$349$	4.04756e6			1.77881			0.889405	+	0.457120i	$0.151119\pi$
$350$		0			0
$351$	77806.1	+	987414.i	0.0337090	+	0.427791i
$352$		0			0
$353$		−	2.09516e6i		−	0.894913i	−0.894306	−	0.447457i	$-0.852330\pi$
$353$		−	2.09516e6i		−	0.894913i	0.894306	−	0.447457i	$-0.147670\pi$
$354$		0			0
$355$		−	4.31051e6i		−	1.81534i
$356$		0			0
$357$	−1.38213e6	−	750381.i	−0.573958	−	0.311610i
$358$		0			0
$359$	−4.66958e6			−1.91224			−0.956119	−	0.292979i	$-0.905353\pi$
$359$	−4.66958e6			−1.91224			−0.956119	+	0.292979i	$0.905353\pi$
$360$		0			0
$361$	1.51899e6			0.613463
$362$		0			0
$363$	−2.15058e6	−	1.16758e6i	−0.856620	−	0.465071i
$364$		0			0
$365$			125550.i			0.0493270i
$366$		0			0
$367$			135444.i			0.0524921i	0.999656	+	0.0262461i	$0.00835534\pi$
$367$			135444.i			0.0524921i	−0.999656	+	0.0262461i	$0.991645\pi$
$368$		0			0
$369$	−3.60779e6	+	2.34325e6i	−1.37935	+	0.895887i
$370$		0			0
$371$	1.21744e6			0.459213
$372$		0			0
$373$	4.02024e6			1.49617			0.748083	−	0.663605i	$-0.230975\pi$
$373$	4.02024e6			1.49617			0.748083	+	0.663605i	$0.230975\pi$
$374$		0			0
$375$	3.01061e6	−	5.54527e6i	1.10554	−	2.03631i
$376$		0			0
$377$		−	1.46544e6i		−	0.531023i
$378$		0			0
$379$		−	4.14968e6i		−	1.48394i	−0.670433	−	0.741970i	$-0.733892\pi$
$379$		−	4.14968e6i		−	1.48394i	0.670433	−	0.741970i	$-0.266108\pi$
$380$		0			0
$381$	1.34065e6	−	2.46936e6i	0.473154	−	0.871508i
$382$		0			0
$383$	1.10473e6			0.384823			0.192411	−	0.981314i	$-0.438369\pi$
$383$	1.10473e6			0.384823			0.192411	+	0.981314i	$0.438369\pi$
$384$		0			0
$385$	−2.79743e6			−0.961850
$386$		0			0
$387$	2.51253e6	−	1.63188e6i	0.852773	−	0.553875i
$388$		0			0
$389$		−	5.71601e6i		−	1.91522i	−0.288065	−	0.957611i	$-0.593012\pi$
$389$		−	5.71601e6i		−	1.91522i	0.288065	−	0.957611i	$-0.406988\pi$
$390$		0			0
$391$			4.54742e6i			1.50426i
$392$		0			0
$393$	1.10601e6	+	600470.i	0.361226	+	0.196115i
$394$		0			0
$395$	6.40158e6			2.06440
$396$		0			0
$397$	2.81029e6			0.894900			0.447450	−	0.894309i	$-0.352332\pi$
$397$	2.81029e6			0.894900			0.447450	+	0.894309i	$0.352332\pi$
$398$		0			0
$399$	−656727.	−	356547.i	−0.206515	−	0.112120i
$400$		0			0
$401$			3.08108e6i			0.956847i	0.878129	+	0.478423i	$0.158792\pi$
$401$			3.08108e6i			0.956847i	−0.878129	+	0.478423i	$0.841208\pi$
$402$		0			0
$403$		−	1.23867e6i		−	0.379921i
$404$		0			0
$405$	5.46129e6	+	2.43069e6i	1.65447	+	0.736363i
$406$		0			0
$407$	6.10028e6			1.82542
$408$		0			0
$409$	2.31720e6			0.684944			0.342472	−	0.939528i	$-0.388736\pi$
$409$	2.31720e6			0.684944			0.342472	+	0.939528i	$0.388736\pi$
$410$		0			0
$411$	−1.21073e6	+	2.23006e6i	−0.353544	+	0.651196i
$412$		0			0
$413$			1.04876e6i			0.302553i
$414$		0			0
$415$		−	6.29960e6i		−	1.79553i
$416$		0			0
$417$	−670227.	+	1.23450e6i	−0.188748	+	0.347657i
$418$		0			0
$419$	4.48478e6			1.24798			0.623988	−	0.781434i	$-0.285511\pi$
$419$	4.48478e6			1.24798			0.623988	+	0.781434i	$0.285511\pi$
$420$		0			0
$421$	146670.			0.0403307			0.0201653	−	0.999797i	$-0.493581\pi$
$421$	146670.			0.0403307			0.0201653	+	0.999797i	$0.493581\pi$
$422$		0			0
$423$	−887039.	−	1.36573e6i	−0.241041	−	0.371119i
$424$		0			0
$425$		−	1.46667e7i		−	3.93876i
$426$		0			0
$427$		−	230174.i		−	0.0610924i
$428$		0			0
$429$	2.02012e6	+	1.09675e6i	0.529950	+	0.287717i
$430$		0			0
$431$	−1.57335e6			−0.407974			−0.203987	−	0.978974i	$-0.565390\pi$
$431$	−1.57335e6			−0.407974			−0.203987	+	0.978974i	$0.565390\pi$
$432$		0			0
$433$	2.73055e6			0.699891			0.349945	−	0.936770i	$-0.386200\pi$
$433$	2.73055e6			0.699891			0.349945	+	0.936770i	$0.386200\pi$
$434$		0			0
$435$	−7.77267e6	−	4.21989e6i	−1.96946	−	1.06925i
$436$		0			0
$437$			2.16073e6i			0.541248i
$438$		0			0
$439$		−	937465.i		−	0.232163i	−0.993240	−	0.116082i	$-0.962967\pi$
$439$		−	937465.i		−	0.232163i	0.993240	−	0.116082i	$-0.0370334\pi$
$440$		0			0
$441$	−317797.	−	489296.i	−0.0778132	−	0.119805i
$442$		0			0
$443$	6.42837e6			1.55629			0.778147	−	0.628082i	$-0.216160\pi$
$443$	6.42837e6			1.55629			0.778147	+	0.628082i	$0.216160\pi$
$444$		0			0
$445$	2.78841e6			0.667508
$446$		0			0
$447$	−2.42988e6	+	4.47562e6i	−0.575196	+	1.05946i
$448$		0			0
$449$			5.18414e6i			1.21356i	0.794870	+	0.606780i	$0.207539\pi$
$449$			5.18414e6i			1.21356i	−0.794870	+	0.606780i	$0.792461\pi$
$450$		0			0
$451$			9.98382e6i			2.31129i
$452$		0			0
$453$	−2.36062e6	+	4.34806e6i	−0.540482	+	0.995520i
$454$		0			0
$455$	1.29705e6			0.293717
$456$		0			0
$457$	2.22298e6			0.497903			0.248952	−	0.968516i	$-0.419914\pi$
$457$	2.22298e6			0.497903			0.248952	+	0.968516i	$0.419914\pi$
$458$		0			0
$459$	7.77518e6	−	612668.i	1.72258	−	0.135735i
$460$		0			0
$461$			7.79168e6i			1.70757i	0.520625	+	0.853785i	$0.325699\pi$
$461$			7.79168e6i			1.70757i	−0.520625	+	0.853785i	$0.674301\pi$
$462$		0			0
$463$			6.03074e6i			1.30743i	0.756741	+	0.653714i	$0.226790\pi$
$463$			6.03074e6i			1.30743i	−0.756741	+	0.653714i	$0.773210\pi$
$464$		0			0
$465$	−6.56990e6	−	3.56689e6i	−1.40905	−	0.764993i
$466$		0			0
$467$	2.48458e6			0.527182			0.263591	−	0.964635i	$-0.415093\pi$
$467$	2.48458e6			0.527182			0.263591	+	0.964635i	$0.415093\pi$
$468$		0			0
$469$	2.41562e6			0.507104
$470$		0			0
$471$	−1.08559e6	−	589382.i	−0.225483	−	0.122418i
$472$		0			0
$473$		−	6.95291e6i		−	1.42894i
$474$		0			0
$475$		−	6.96894e6i		−	1.41721i
$476$		0			0
$477$	−5.06329e6	+	3.28860e6i	−1.01891	+	0.661782i
$478$		0			0
$479$	3.85256e6			0.767203			0.383601	−	0.923499i	$-0.374684\pi$
$479$	3.85256e6			0.767203			0.383601	+	0.923499i	$0.374684\pi$
$480$		0			0
$481$	−2.82845e6			−0.557424
$482$		0			0
$483$	−804927.	+	1.48260e6i	−0.156996	+	0.289173i
$484$		0			0
$485$			1.80820e7i			3.49054i
$486$		0			0
$487$		−	2.30416e6i		−	0.440241i	−0.975473	−	0.220120i	$-0.929355\pi$
$487$		−	2.30416e6i		−	0.440241i	0.975473	−	0.220120i	$-0.0706450\pi$
$488$		0			0
$489$	453429.	−	835177.i	0.0857507	−	0.157945i
$490$		0			0
$491$	7.47471e6			1.39923			0.699617	−	0.714518i	$-0.253354\pi$
$491$	7.47471e6			1.39923			0.699617	+	0.714518i	$0.253354\pi$
$492$		0			0
$493$	−1.15393e7			−2.13826
$494$		0			0
$495$	1.16344e7	−	7.55650e6i	2.13417	−	1.38614i
$496$		0			0
$497$		−	2.08640e6i		−	0.378884i
$498$		0			0
$499$			5.87745e6i			1.05667i	0.849037	+	0.528333i	$0.177183\pi$
$499$			5.87745e6i			1.05667i	−0.849037	+	0.528333i	$0.822817\pi$
$500$		0			0
$501$	−3.41104e6	−	1.85190e6i	−0.607144	−	0.329627i
$502$		0			0
$503$	−3.42586e6			−0.603739			−0.301870	−	0.953349i	$-0.597611\pi$
$503$	−3.42586e6			−0.603739			−0.301870	+	0.953349i	$0.597611\pi$
$504$		0			0
$505$	5.70660e6			0.995748
$506$		0			0
$507$	4.14993e6	+	2.25306e6i	0.717003	+	0.389271i
$508$		0			0
$509$			1.92843e6i			0.329921i	0.986300	+	0.164961i	$0.0527497\pi$
$509$			1.92843e6i			0.329921i	−0.986300	+	0.164961i	$0.947250\pi$
$510$		0			0
$511$			60769.4i			0.0102952i
$512$		0			0
$513$	3.69441e6	−	291112.i	0.619800	−	0.0488389i
$514$		0			0
$515$	−1.63997e7			−2.72470
$516$		0			0
$517$	−3.77938e6			−0.621862
$518$		0			0
$519$	−79072.0	+	145644.i	−0.0128856	+	0.0237341i
$520$		0			0
$521$			7.54800e6i			1.21825i	0.793073	+	0.609126i	$0.208480\pi$
$521$			7.54800e6i			1.21825i	−0.793073	+	0.609126i	$0.791520\pi$
$522$		0			0
$523$			6.71701e6i			1.07380i	0.843647	+	0.536898i	$0.180404\pi$
$523$			6.71701e6i			1.07380i	−0.843647	+	0.536898i	$0.819596\pi$
$524$		0			0
$525$	2.59611e6	−	4.78181e6i	0.411079	−	0.757171i
$526$		0			0
$527$	−9.75364e6			−1.52982
$528$		0			0
$529$	−1.55836e6			−0.242119
$530$		0			0
$531$	−2.83295e6	−	4.36175e6i	−0.436016	−	0.671311i
$532$		0			0
$533$		−	4.62909e6i		−	0.705793i
$534$		0			0
$535$		−	2.01297e7i		−	3.04056i
$536$		0			0
$537$	7.52172e6	+	4.08365e6i	1.12559	+	0.611101i
$538$		0			0
$539$	−1.35403e6			−0.200750
$540$		0			0
$541$	6.34084e6			0.931437			0.465719	−	0.884933i	$-0.345796\pi$
$541$	6.34084e6			0.931437			0.465719	+	0.884933i	$0.345796\pi$
$542$		0			0
$543$	1.07474e7	+	5.83491e6i	1.56424	+	0.849248i
$544$		0			0
$545$			2.53899e6i			0.366159i
$546$		0			0
$547$		−	9.65227e6i		−	1.37931i	−0.724139	−	0.689654i	$-0.757763\pi$
$547$		−	9.65227e6i		−	1.37931i	0.724139	−	0.689654i	$-0.242237\pi$
$548$		0			0
$549$	621754.	+	957283.i	0.0880415	+	0.135553i
$550$		0			0
$551$	−5.48293e6			−0.769367
$552$		0			0
$553$	3.09853e6			0.430866
$554$		0			0
$555$	−8.14485e6	+	1.50021e7i	−1.12241	+	2.06737i
$556$		0			0
$557$			8.77938e6i			1.19902i	0.800368	+	0.599509i	$0.204638\pi$
$557$			8.77938e6i			1.19902i	−0.800368	+	0.599509i	$0.795362\pi$
$558$		0			0
$559$			3.22378e6i			0.436351i
$560$		0			0
$561$	8.63616e6	−	1.59070e7i	1.15855	−	2.13394i
$562$		0			0
$563$	1.24019e7			1.64899			0.824497	−	0.565867i	$-0.191458\pi$
$563$	1.24019e7			1.64899			0.824497	+	0.565867i	$0.191458\pi$
$564$		0			0
$565$	−382118.			−0.0503589
$566$		0			0
$567$	2.64340e6	+	1.17652e6i	0.345307	+	0.153688i
$568$		0			0
$569$			6.99633e6i			0.905920i	0.891531	+	0.452960i	$0.149632\pi$
$569$			6.99633e6i			0.905920i	−0.891531	+	0.452960i	$0.850368\pi$
$570$		0			0
$571$		−	1.15714e7i		−	1.48523i	−0.669717	−	0.742616i	$-0.733585\pi$
$571$		−	1.15714e7i		−	1.48523i	0.669717	−	0.742616i	$-0.266415\pi$
$572$		0			0
$573$	−2.16721e6	−	1.17661e6i	−0.275749	−	0.149708i
$574$		0			0
$575$	−1.57328e7			−1.98444
$576$		0			0
$577$	−1.38535e7			−1.73229			−0.866144	−	0.499794i	$-0.833409\pi$
$577$	−1.38535e7			−1.73229			−0.866144	+	0.499794i	$0.833409\pi$
$578$		0			0
$579$	2.75191e6	+	1.49405e6i	0.341144	+	0.185212i
$580$		0			0
$581$		−	3.04917e6i		−	0.374749i
$582$		0			0
$583$			1.40116e7i			1.70733i
$584$		0			0
$585$	−5.39438e6	+	3.50364e6i	−0.651706	+	0.423282i
$586$		0			0
$587$	−8.66075e6			−1.03743			−0.518717	−	0.854946i	$-0.673590\pi$
$587$	−8.66075e6			−1.03743			−0.518717	+	0.854946i	$0.673590\pi$
$588$		0			0
$589$	−4.63448e6			−0.550444
$590$		0			0
$591$	4.97946e6	−	9.17173e6i	0.586427	−	1.08015i
$592$		0			0
$593$		−	1.51120e7i		−	1.76476i	−0.470537	−	0.882380i	$-0.655940\pi$
$593$		−	1.51120e7i		−	1.76476i	0.470537	−	0.882380i	$-0.344060\pi$
$594$		0			0
$595$		−	1.02134e7i		−	1.18271i
$596$		0			0
$597$	−987267.	+	1.81846e6i	−0.113370	+	0.208818i
$598$		0			0
$599$	−8.30546e6			−0.945794			−0.472897	−	0.881118i	$-0.656792\pi$
$599$	−8.30546e6			−0.945794			−0.472897	+	0.881118i	$0.656792\pi$
$600$		0			0
$601$	−6.18429e6			−0.698399			−0.349200	−	0.937048i	$-0.613546\pi$
$601$	−6.18429e6			−0.698399			−0.349200	+	0.937048i	$0.613546\pi$
$602$		0			0
$603$	−1.00464e7	+	6.52515e6i	−1.12517	+	0.730798i
$604$		0			0
$605$		−	1.58918e7i		−	1.76516i
$606$		0			0
$607$			5.81189e6i			0.640244i	0.947376	+	0.320122i	$0.103724\pi$
$607$			5.81189e6i			0.640244i	−0.947376	+	0.320122i	$0.896276\pi$
$608$		0			0
$609$	−3.76217e6	−	2.04254e6i	−0.411050	−	0.223165i
$610$		0			0
$611$	1.75234e6			0.189896
$612$		0			0
$613$	−1.07830e7			−1.15901			−0.579505	−	0.814969i	$-0.696754\pi$
$613$	−1.07830e7			−1.15901			−0.579505	+	0.814969i	$0.696754\pi$
$614$		0			0
$615$	−2.45527e7	−	1.33300e7i	−2.61765	−	1.42116i
$616$		0			0
$617$		−	2.43281e6i		−	0.257274i	−0.991692	−	0.128637i	$-0.958940\pi$
$617$		−	2.43281e6i		−	0.257274i	0.991692	−	0.128637i	$-0.0410601\pi$
$618$		0			0
$619$			8.09525e6i			0.849187i	0.905384	+	0.424594i	$0.139583\pi$
$619$			8.09525e6i			0.849187i	−0.905384	+	0.424594i	$0.860417\pi$
$620$		0			0
$621$	−657203.	−	8.34037e6i	−0.0683866	−	0.867873i
$622$		0			0
$623$	1.34966e6			0.139317
$624$		0			0
$625$	1.87165e7			1.91657
$626$		0			0
$627$	4.10351e6	−	7.55829e6i	0.416856	−	0.767812i
$628$		0			0
$629$			2.22720e7i			2.24457i
$630$		0			0
$631$			8140.59i			0.000813921i	1.00000		0.000406961i	$0.000129540\pi$
$631$			8140.59i			0.000813921i	−1.00000		0.000406961i	$0.999870\pi$
$632$		0			0
$633$	2.27070e6	−	4.18243e6i	0.225243	−	0.414877i
$634$		0			0
$635$	1.82475e7			1.79584
$636$		0			0
$637$	627807.			0.0613024
$638$		0			0
$639$	5.63584e6	+	8.67722e6i	0.546017	+	0.840676i
$640$		0			0
$641$		−	1.11109e7i		−	1.06808i	−0.845459	−	0.534040i	$-0.820673\pi$
$641$		−	1.11109e7i		−	1.06808i	0.845459	−	0.534040i	$-0.179327\pi$
$642$		0			0
$643$		−	3.45574e6i		−	0.329621i	−0.986325	−	0.164810i	$-0.947299\pi$
$643$		−	3.45574e6i		−	0.329621i	0.986325	−	0.164810i	$-0.0527012\pi$
$644$		0			0
$645$	1.70989e7	+	9.28325e6i	1.61834	+	0.878619i
$646$		0			0
$647$	−1.79213e6			−0.168309			−0.0841546	−	0.996453i	$-0.526819\pi$
$647$	−1.79213e6			−0.168309			−0.0841546	+	0.996453i	$0.526819\pi$
$648$		0			0
$649$	−1.20702e7			−1.12487
$650$		0			0
$651$	−3.18000e6	−	1.72647e6i	−0.294086	−	0.159664i
$652$		0			0
$653$		−	4.92512e6i		−	0.451995i	−0.974128	−	0.225997i	$-0.927436\pi$
$653$		−	4.92512e6i		−	0.451995i	0.974128	−	0.225997i	$-0.0725641\pi$
$654$		0			0
$655$			8.17295e6i			0.744347i
$656$		0			0
$657$	−164152.	−	252737.i	−0.0148366	−	0.0228431i
$658$		0			0
$659$	−1.32977e7			−1.19278			−0.596392	−	0.802693i	$-0.703400\pi$
$659$	−1.32977e7			−1.19278			−0.596392	+	0.802693i	$0.703400\pi$
$660$		0			0
$661$	1.34181e7			1.19451			0.597253	−	0.802053i	$-0.296259\pi$
$661$	1.34181e7			1.19451			0.597253	+	0.802053i	$0.296259\pi$
$662$		0			0
$663$	−4.00424e6	+	7.37544e6i	−0.353782	+	0.651635i
$664$		0			0
$665$		−	4.85293e6i		−	0.425549i
$666$		0			0
$667$			1.23781e7i			1.07730i
$668$		0			0
$669$	5.80248e6	−	1.06876e7i	0.501243	−	0.923245i
$670$		0			0
$671$	2.64908e6			0.227138
$672$		0			0
$673$	1.23185e7			1.04838			0.524191	−	0.851601i	$-0.324368\pi$
$673$	1.23185e7			1.04838			0.524191	+	0.851601i	$0.324368\pi$
$674$		0			0
$675$	2.11966e6	+	2.69000e7i	0.179064	+	2.27244i
$676$		0			0
$677$		−	1.08730e7i		−	0.911758i	−0.890042	−	0.455879i	$-0.849325\pi$
$677$		−	1.08730e7i		−	0.911758i	0.890042	−	0.455879i	$-0.150675\pi$
$678$		0			0
$679$			8.75215e6i			0.728518i
$680$		0			0
$681$	9.40225e6	+	5.10462e6i	0.776898	+	0.421789i
$682$		0			0
$683$	6.87074e6			0.563575			0.281788	−	0.959477i	$-0.409073\pi$
$683$	6.87074e6			0.563575			0.281788	+	0.959477i	$0.409073\pi$
$684$		0			0
$685$	−1.64792e7			−1.34187
$686$		0			0
$687$	8.49414e6	+	4.61159e6i	0.686638	+	0.372786i
$688$		0			0
$689$		−	6.49661e6i		−	0.521361i
$690$		0			0
$691$		−	3.01872e6i		−	0.240507i	−0.992743	−	0.120254i	$-0.961629\pi$
$691$		−	3.01872e6i		−	0.240507i	0.992743	−	0.120254i	$-0.0383708\pi$
$692$		0			0
$693$	5.63133e6	−	3.65754e6i	0.445428	−	0.289305i
$694$		0			0
$695$	−9.12241e6			−0.716387
$696$		0			0
$697$	−3.64508e7			−2.84201
$698$		0			0
$699$	−3.34449e6	+	6.16026e6i	−0.258903	+	0.476877i
$700$		0			0
$701$			5.85356e6i			0.449909i	0.974369	+	0.224955i	$0.0722234\pi$
$701$			5.85356e6i			0.449909i	−0.974369	+	0.224955i	$0.927777\pi$
$702$		0			0
$703$			1.05826e7i			0.807618i
$704$		0			0
$705$	5.04607e6	−	9.29442e6i	0.382367	−	0.704286i
$706$		0			0
$707$	2.76214e6			0.207825
$708$		0			0
$709$	−3.30620e6			−0.247009			−0.123505	−	0.992344i	$-0.539413\pi$
$709$	−3.30620e6			−0.247009			−0.123505	+	0.992344i	$0.539413\pi$
$710$		0			0
$711$	−1.28866e7	+	8.36984e6i	−0.956016	+	0.620931i
$712$		0			0
$713$			1.04626e7i			0.770758i
$714$		0			0
$715$			1.49278e7i			1.09202i
$716$		0			0
$717$	−4.08944e6	−	2.22022e6i	−0.297075	−	0.161286i
$718$		0			0
$719$	−1.08841e7			−0.785182			−0.392591	−	0.919713i	$-0.628421\pi$
$719$	−1.08841e7			−0.785182			−0.392591	+	0.919713i	$0.628421\pi$
$720$		0			0
$721$	−7.93789e6			−0.568679
$722$		0			0
$723$	−7.90525e6	−	4.29187e6i	−0.562432	−	0.305352i
$724$		0			0
$725$		−	3.99227e7i		−	2.82082i
$726$		0			0
$727$		−	2.21731e6i		−	0.155593i	−0.996969	−	0.0777967i	$-0.975212\pi$
$727$		−	2.21731e6i		−	0.155593i	0.996969	−	0.0777967i	$-0.0247885\pi$
$728$		0			0
$729$	−1.41718e7	+	2.24737e6i	−0.987658	+	0.156623i
$730$		0			0
$731$	2.53850e7			1.75705
$732$		0			0
$733$	−1.46059e7			−1.00408			−0.502041	−	0.864844i	$-0.667417\pi$
$733$	−1.46059e7			−1.00408			−0.502041	+	0.864844i	$0.667417\pi$
$734$		0			0
$735$	1.80784e6	−	3.32989e6i	0.123436	−	0.227358i
$736$		0			0
$737$			2.78015e7i			1.88538i
$738$		0			0
$739$		−	2.10866e7i		−	1.42035i	−0.704025	−	0.710175i	$-0.748616\pi$
$739$		−	2.10866e7i		−	1.42035i	0.704025	−	0.710175i	$-0.251384\pi$
$740$		0			0
$741$	−1.90263e6	+	3.50447e6i	−0.127294	+	0.234464i
$742$		0			0
$743$	6.99253e6			0.464689			0.232345	−	0.972634i	$-0.425360\pi$
$743$	6.99253e6			0.464689			0.232345	+	0.972634i	$0.425360\pi$
$744$		0			0
$745$	−3.30729e7			−2.18314
$746$		0			0
$747$	8.23650e6	+	1.26813e7i	0.540059	+	0.831502i
$748$		0			0
$749$		−	9.74329e6i		−	0.634601i
$750$		0			0
$751$		−	1.84176e7i		−	1.19161i	−0.803131	−	0.595803i	$-0.796834\pi$
$751$		−	1.84176e7i		−	1.19161i	0.803131	−	0.595803i	$-0.203166\pi$
$752$		0			0
$753$	1.48602e7	+	8.06779e6i	0.955072	+	0.518522i
$754$		0			0
$755$	−3.21303e7			−2.05138
$756$		0			0
$757$	2.56103e7			1.62434			0.812168	−	0.583424i	$-0.198287\pi$
$757$	2.56103e7			1.62434			0.812168	+	0.583424i	$0.198287\pi$
$758$		0			0
$759$	−1.70633e7	−	9.26394e6i	−1.07513	−	0.583702i
$760$		0			0
$761$		−	6.79815e6i		−	0.425529i	−0.977104	−	0.212764i	$-0.931753\pi$
$761$		−	6.79815e6i		−	0.425529i	0.977104	−	0.212764i	$-0.0682467\pi$
$762$		0			0
$763$			1.22894e6i			0.0764220i
$764$		0			0
$765$	2.75887e7	+	4.24769e7i	1.70442	+	2.62421i
$766$		0			0
$767$	5.59647e6			0.343499
$768$		0			0
$769$	−7.38515e6			−0.450343			−0.225171	−	0.974319i	$-0.572294\pi$
$769$	−7.38515e6			−0.450343			−0.225171	+	0.974319i	$0.572294\pi$
$770$		0			0
$771$	−5.47234e6	+	1.00796e7i	−0.331541	+	0.610669i
$772$		0			0
$773$		−	2.06189e7i		−	1.24113i	−0.784154	−	0.620566i	$-0.786903\pi$
$773$		−	2.06189e7i		−	1.24113i	0.784154	−	0.620566i	$-0.213097\pi$
$774$		0			0
$775$		−	3.37449e7i		−	2.01815i
$776$		0			0
$777$	−3.94231e6	+	7.26139e6i	−0.234260	+	0.431486i
$778$		0			0
$779$	−1.73197e7			−1.02258
$780$		0			0
$781$	2.40124e7			1.40867
$782$		0			0
$783$	2.11640e7	−	1.66768e6i	1.23366	−	0.0972094i
$784$		0			0
$785$		−	8.02203e6i		−	0.464633i
$786$		0			0
$787$		−	5.73022e6i		−	0.329788i	−0.986311	−	0.164894i	$-0.947272\pi$
$787$		−	5.73022e6i		−	0.329788i	0.986311	−	0.164894i	$-0.0527282\pi$
$788$		0			0
$789$	−9.97346e6	−	5.41473e6i	−0.570365	−	0.309660i
$790$		0			0
$791$	−184955.			−0.0105105
$792$		0			0
$793$	−1.22827e6			−0.0693604
$794$		0			0
$795$	−3.44580e7	−	1.87077e7i	−1.93362	−	1.04979i
$796$		0			0
$797$		−	2.28573e7i		−	1.27461i	−0.770610	−	0.637307i	$-0.780048\pi$
$797$		−	2.28573e7i		−	1.27461i	0.770610	−	0.637307i	$-0.219952\pi$
$798$		0			0
$799$		−	1.37984e7i		−	0.764651i
$800$		0			0
$801$	−5.61317e6	+	3.64574e6i	−0.309120	+	0.200773i
$802$		0			0
$803$	−699397.			−0.0382767
$804$		0			0
$805$	−1.09558e7			−0.595874
$806$		0			0
$807$	2.76592e6	−	5.09459e6i	0.149505	−	0.275375i
$808$		0			0
$809$			2.69299e7i			1.44665i	0.690507	+	0.723326i	$0.257387\pi$
$809$			2.69299e7i			1.44665i	−0.690507	+	0.723326i	$0.742613\pi$
$810$		0			0
$811$			8.40782e6i			0.448881i	0.974488	+	0.224441i	$0.0720555\pi$
$811$			8.40782e6i			0.448881i	−0.974488	+	0.224441i	$0.927945\pi$
$812$		0			0
$813$	−3.00006e6	+	5.52585e6i	−0.159186	+	0.293206i
$814$		0			0
$815$	6.17159e6			0.325464
$816$		0			0
$817$	1.20618e7			0.632202
$818$		0			0
$819$	−2.61102e6	+	1.69585e6i	−0.136019	+	0.0883441i
$820$		0			0
$821$		−	1.00689e7i		−	0.521343i	−0.965428	−	0.260671i	$-0.916056\pi$
$821$		−	1.00689e7i		−	0.521343i	0.965428	−	0.260671i	$-0.0839439\pi$
$822$		0			0
$823$			3.14676e7i			1.61943i	0.586820	+	0.809717i	$0.300380\pi$
$823$			3.14676e7i			1.61943i	−0.586820	+	0.809717i	$0.699620\pi$
$824$		0			0
$825$	5.50340e7	+	2.98788e7i	2.81512	+	1.52837i
$826$		0			0
$827$	−8.46941e6			−0.430615			−0.215308	−	0.976546i	$-0.569075\pi$
$827$	−8.46941e6			−0.430615			−0.215308	+	0.976546i	$0.569075\pi$
$828$		0			0
$829$	−2.57185e7			−1.29975			−0.649873	−	0.760042i	$-0.725178\pi$
$829$	−2.57185e7			−1.29975			−0.649873	+	0.760042i	$0.725178\pi$
$830$		0			0
$831$	−3.76339e6	−	2.04320e6i	−0.189050	−	0.102638i
$832$		0			0
$833$		−	4.94353e6i		−	0.246845i
$834$		0			0
$835$		−	2.52061e7i		−	1.25109i
$836$		0			0
$837$	1.78890e7	−	1.40962e6i	0.882619	−	0.0695485i
$838$		0			0
$839$	1.54091e7			0.755740			0.377870	−	0.925859i	$-0.376657\pi$
$839$	1.54091e7			0.755740			0.377870	+	0.925859i	$0.376657\pi$
$840$		0			0
$841$	−1.08987e7			−0.531356
$842$		0			0
$843$	9.43244e6	−	1.73737e7i	0.457146	−	0.842022i
$844$		0			0
$845$			3.06662e7i			1.47747i
$846$		0			0
$847$		−	7.69204e6i		−	0.368412i
$848$		0			0
$849$	−6.15151e6	+	1.13305e7i	−0.292895	+	0.539487i
$850$		0			0
$851$	2.38910e7			1.13086
$852$		0			0
$853$	2.27521e7			1.07065			0.535326	−	0.844646i	$-0.320189\pi$
$853$	2.27521e7			1.07065			0.535326	+	0.844646i	$0.320189\pi$
$854$		0			0
$855$	1.31089e7	+	2.01831e7i	0.613267	+	0.944217i
$856$		0			0
$857$			3.88221e6i			0.180562i	0.995916	+	0.0902811i	$0.0287765\pi$
$857$			3.88221e6i			0.180562i	−0.995916	+	0.0902811i	$0.971223\pi$
$858$		0			0
$859$		−	4.16792e7i		−	1.92724i	−0.267272	−	0.963621i	$-0.586122\pi$
$859$		−	4.16792e7i		−	1.92724i	0.267272	−	0.963621i	$-0.413878\pi$
$860$		0			0
$861$	−1.18841e7	−	6.45206e6i	−0.546335	−	0.296613i
$862$		0			0
$863$	−3.09780e7			−1.41588			−0.707939	−	0.706273i	$-0.750375\pi$
$863$	−3.09780e7			−1.41588			−0.707939	+	0.706273i	$0.750375\pi$
$864$		0			0
$865$	−1.07624e6			−0.0489069
$866$		0			0
$867$	3.86249e7	+	2.09700e7i	1.74510	+	0.947437i
$868$		0			0
$869$			3.56611e7i			1.60194i
$870$		0			0
$871$		−	1.28904e7i		−	0.575733i
$872$		0			0
$873$	−2.36416e7	−	3.63997e7i	−1.04988	−	1.61645i
$874$		0			0
$875$	1.98340e7			0.875769
$876$		0			0
$877$	−1.15935e7			−0.508996			−0.254498	−	0.967073i	$-0.581910\pi$
$877$	−1.15935e7			−0.508996			−0.254498	+	0.967073i	$0.581910\pi$
$878$		0			0
$879$	3.89270e6	−	7.17001e6i	0.169934	−	0.313002i
$880$		0			0
$881$		−	8.98792e6i		−	0.390139i	−0.980789	−	0.195069i	$-0.937507\pi$
$881$		−	8.98792e6i		−	0.390139i	0.980789	−	0.195069i	$-0.0624932\pi$
$882$		0			0
$883$		−	4.01117e7i		−	1.73129i	−0.500658	−	0.865645i	$-0.666909\pi$
$883$		−	4.01117e7i		−	1.73129i	0.500658	−	0.865645i	$-0.333091\pi$
$884$		0			0
$885$	1.61157e7	−	2.96837e7i	0.691657	−	1.27397i
$886$		0			0
$887$	2.17174e7			0.926827			0.463413	−	0.886142i	$-0.346624\pi$
$887$	2.17174e7			0.926827			0.463413	+	0.886142i	$0.346624\pi$
$888$		0			0
$889$	8.83224e6			0.374814
$890$		0			0
$891$	−1.35406e7	+	3.04230e7i	−0.571403	+	1.28383i
$892$		0			0
$893$		−	6.55639e6i		−	0.275129i
$894$		0			0
$895$			5.55822e7i			2.31941i
$896$		0			0
$897$	7.91158e6	+	4.29531e6i	0.328308	+	0.178243i
$898$		0			0
$899$	−2.65494e7			−1.09561
$900$		0			0
$901$	−5.11562e7			−2.09936
$902$		0			0
$903$	8.27631e6	+	4.49333e6i	0.337767	+	0.183379i
$904$		0			0
$905$			7.94185e7i			3.22329i
$906$		0			0
$907$		−	2.58097e7i		−	1.04175i	−0.853632	−	0.520877i	$-0.825605\pi$
$907$		−	2.58097e7i		−	1.04175i	0.853632	−	0.520877i	$-0.174395\pi$
$908$		0			0
$909$	−1.14876e7	+	7.46118e6i	−0.461126	+	0.299501i
$910$		0			0
$911$	8.88878e6			0.354851			0.177426	−	0.984134i	$-0.443223\pi$
$911$	8.88878e6			0.354851			0.177426	+	0.984134i	$0.443223\pi$
$912$		0			0
$913$	3.50930e7			1.39329
$914$		0			0
$915$	−3.53695e6	+	6.51475e6i	−0.139661	+	0.257244i
$916$		0			0
$917$			3.95592e6i			0.155354i
$918$		0			0
$919$			4.58880e7i			1.79230i	0.443754	+	0.896149i	$0.353646\pi$
$919$			4.58880e7i			1.79230i	−0.443754	+	0.896149i	$0.646354\pi$
$920$		0			0
$921$	1.15991e7	−	2.13644e7i	0.450582	−	0.829932i
$922$		0			0
$923$	−1.11336e7			−0.430161
$924$		0			0
$925$	−7.70551e7			−2.96106
$926$		0			0
$927$	3.30133e7	−	2.14421e7i	1.26180	−	0.819535i
$928$		0			0
$929$			9.04074e6i			0.343688i	0.985124	+	0.171844i	$0.0549725\pi$
$929$			9.04074e6i			0.343688i	−0.985124	+	0.171844i	$0.945027\pi$
$930$		0			0
$931$		−	2.34894e6i		−	0.0888173i
$932$		0			0
$933$	−1.44241e7	−	7.83103e6i	−0.542480	−	0.294520i
$934$		0			0
$935$	1.17546e8			4.39723
$936$		0			0
$937$	4.77522e7			1.77683			0.888413	−	0.459046i	$-0.151809\pi$
$937$	4.77522e7			1.77683			0.888413	+	0.459046i	$0.151809\pi$
$938$		0			0
$939$	8.51763e6	+	4.62434e6i	0.315250	+	0.171154i
$940$		0			0
$941$		−	5.67795e6i		−	0.209034i	−0.994523	−	0.104517i	$-0.966670\pi$
$941$		−	5.67795e6i		−	0.209034i	0.994523	−	0.104517i	$-0.0333297\pi$
$942$		0			0
$943$			3.91004e7i			1.43187i
$944$		0			0
$945$	1.47606e6	+	1.87322e7i	0.0537681	+	0.682354i
$946$		0			0
$947$	−3.09672e6			−0.112209			−0.0561044	−	0.998425i	$-0.517868\pi$
$947$	−3.09672e6			−0.112209			−0.0561044	+	0.998425i	$0.517868\pi$
$948$		0			0
$949$	324282.			0.0116885
$950$		0			0
$951$	4.19965e6	−	7.73538e6i	0.150578	−	0.277351i
$952$		0			0
$953$			2.78609e7i			0.993717i	0.867831	+	0.496859i	$0.165513\pi$
$953$			2.78609e7i			0.993717i	−0.867831	+	0.496859i	$0.834487\pi$
$954$		0			0
$955$		−	1.60147e7i		−	0.568214i
$956$		0			0
$957$	2.35076e7	−	4.32989e7i	0.829715	−	1.52826i
$958$		0			0
$959$	−7.97634e6			−0.280064
$960$		0			0
$961$	6.18810e6			0.216147
$962$		0			0
$963$	2.63189e7	+	4.05218e7i	0.914537	+	1.40807i
$964$		0			0
$965$			2.03354e7i			0.702966i
$966$		0			0
$967$			64790.9i			0.00222817i	0.999999	+	0.00111408i	$0.000354624\pi$
$967$			64790.9i			0.00222817i	−0.999999	+	0.00111408i	$0.999645\pi$
$968$		0			0
$969$	2.75952e7	+	1.49818e7i	0.944114	+	0.512573i
$970$		0			0
$971$	4.20220e7			1.43030			0.715152	−	0.698969i	$-0.246357\pi$
$971$	4.20220e7			1.43030			0.715152	+	0.698969i	$0.246357\pi$
$972$		0			0
$973$	−4.41548e6			−0.149519
$974$		0			0
$975$	−2.55170e7	−	1.38536e7i	−0.859644	−	0.466713i
$976$		0			0
$977$		−	438060.i		−	0.0146824i	−0.999973	−	0.00734121i	$-0.997663\pi$
$977$		−	438060.i		−	0.0146824i	0.999973	−	0.00734121i	$-0.00233680\pi$
$978$		0			0
$979$			1.55333e7i			0.517973i
$980$		0			0
$981$	−3.31964e6	−	5.11109e6i	−0.110133	−	0.169567i
$982$		0			0
$983$	1.33014e7			0.439050			0.219525	−	0.975607i	$-0.429549\pi$
$983$	1.33014e7			0.439050			0.219525	+	0.975607i	$0.429549\pi$
$984$		0			0
$985$	6.77751e7			2.22577
$986$		0			0
$987$	2.44243e6	−	4.49873e6i	0.0798047	−	0.146993i
$988$		0			0
$989$		−	2.72302e7i		−	0.885239i
$990$		0			0
$991$			6.84763e6i			0.221491i	0.993849	+	0.110746i	$0.0353238\pi$
$991$			6.84763e6i			0.221491i	−0.993849	+	0.110746i	$0.964676\pi$
$992$		0			0
$993$	1.96750e7	−	3.62397e7i	0.633202	−	1.16630i
$994$		0			0
$995$	−1.34376e7			−0.430293
$996$		0			0
$997$	2.61215e7			0.832262			0.416131	−	0.909305i	$-0.363386\pi$
$997$	2.61215e7			0.832262			0.416131	+	0.909305i	$0.363386\pi$
$998$		0			0
$999$	−3.21880e6	−	4.08488e7i	−0.102042	−	1.29499i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.b.239.5	✓	40
3.2	odd	2	inner	336.6.h.b.239.35	yes	40
4.3	odd	2	inner	336.6.h.b.239.36	yes	40
12.11	even	2	inner	336.6.h.b.239.6	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.b.239.5	✓	40	1.1	even	1	trivial
336.6.h.b.239.6	yes	40	12.11	even	2	inner
336.6.h.b.239.35	yes	40	3.2	odd	2	inner
336.6.h.b.239.36	yes	40	4.3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-13.6996 - 7.43773i) q^{3} -101.234i q^{5} -49.0000i q^{7} +(132.360 + 203.789i) q^{9} +O(q^{10})\)	\(q+(-13.6996 - 7.43773i) q^{3} -101.234i q^{5} -49.0000i q^{7} +(132.360 + 203.789i) q^{9} +563.943 q^{11} -261.477 q^{13} +(-752.954 + 1386.87i) q^{15} +2058.95i q^{17} +978.317i q^{19} +(-364.449 + 671.282i) q^{21} +2208.61 q^{23} -7123.40 q^{25} +(-297.564 - 3776.29i) q^{27} +5604.45i q^{29} +4737.20i q^{31} +(-7725.81 - 4194.46i) q^{33} -4960.48 q^{35} +10817.2 q^{37} +(3582.14 + 1944.80i) q^{39} +17703.6i q^{41} -12329.1i q^{43} +(20630.4 - 13399.4i) q^{45} -6701.70 q^{47} -2401.00 q^{49} +(15313.9 - 28206.8i) q^{51} +24845.8i q^{53} -57090.4i q^{55} +(7276.46 - 13402.6i) q^{57} -21403.3 q^{59} +4697.43 q^{61} +(9985.64 - 6485.65i) q^{63} +26470.5i q^{65} +49298.4i q^{67} +(-30257.2 - 16427.1i) q^{69} +42579.5 q^{71} -1240.19 q^{73} +(97588.0 + 52981.9i) q^{75} -27633.2i q^{77} +63235.3i q^{79} +(-24010.5 + 53947.0i) q^{81} +62227.9 q^{83} +208436. q^{85} +(41684.4 - 76779.0i) q^{87} +27544.1i q^{89} +12812.4i q^{91} +(35234.0 - 64897.9i) q^{93} +99039.3 q^{95} -178615. q^{97} +(74643.6 + 114925. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more