LMFDB - Embedded newform 208.8.f.a.129.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.129"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 449x^{4} + 37224x^{2} + 205776 $ x^6 + 449x^4 + 37224x^2 + 205776
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.5
Root			$-2.43912i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.8.f.a.129.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+51.4496 q^{3} -488.312i q^{5} +616.243i q^{7} +460.056 q^{9} +4625.43i q^{11} +(1257.79 + 7820.90i) q^{13} -25123.4i q^{15} -19536.2 q^{17} +20077.3i q^{19} +31705.4i q^{21} -71368.5 q^{23} -160323. q^{25} -88850.5 q^{27} +138095. q^{29} -18285.7i q^{31} +237976. i q^{33} +300918. q^{35} +446035. i q^{37} +(64712.5 + 402382. i) q^{39} -101639. i q^{41} -103690. q^{43} -224651. i q^{45} +445129. i q^{47} +443788. q^{49} -1.00513e6 q^{51} +203578. q^{53} +2.25865e6 q^{55} +1.03297e6i q^{57} +2.50176e6i q^{59} -792320. q^{61} +283506. i q^{63} +(3.81904e6 - 614191. i) q^{65} -3.17309e6i q^{67} -3.67188e6 q^{69} +1.85875e6i q^{71} -4.25734e6i q^{73} -8.24856e6 q^{75} -2.85039e6 q^{77} -2.38970e6 q^{79} -5.57746e6 q^{81} -1.12536e6i q^{83} +9.53976e6i q^{85} +7.10495e6 q^{87} +8.00227e6i q^{89} +(-4.81957e6 + 775101. i) q^{91} -940791. i q^{93} +9.80397e6 q^{95} +276982. i q^{97} +2.12796e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 56 q^{3} - 1150 q^{9} - 5018 q^{13} + 13152 q^{17} - 27264 q^{23} - 18262 q^{25} - 194560 q^{27} + 42924 q^{29} + 546720 q^{35} - 511160 q^{39} + 1005576 q^{43} + 3246846 q^{49} - 297984 q^{51} + 1705524 q^{53}+ \cdots + 22075632 q^{95}+O(q^{100}) $ 6 * q + 56 * q^3 - 1150 * q^9 - 5018 * q^13 + 13152 * q^17 - 27264 * q^23 - 18262 * q^25 - 194560 * q^27 + 42924 * q^29 + 546720 * q^35 - 511160 * q^39 + 1005576 * q^43 + 3246846 * q^49 - 297984 * q^51 + 1705524 * q^53 + 5320576 * q^55 - 8237572 * q^61 + 7139028 * q^65 - 6017400 * q^69 - 20426072 * q^75 + 4093560 * q^77 - 15147200 * q^79 - 21318442 * q^81 + 36474240 * q^87 - 21281832 * q^91 + 22075632 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	51.4496			1.10016			0.550082	−	0.835111i	$-0.314597\pi$
$3$	51.4496			1.10016			0.550082	+	0.835111i	$0.314597\pi$
$4$		0			0
$5$		−	488.312i		−	1.74704i	−0.486791	−	0.873518i	$-0.661833\pi$
$5$		−	488.312i		−	1.74704i	0.486791	−	0.873518i	$-0.338167\pi$
$6$		0			0
$7$			616.243i			0.679061i	0.940595	+	0.339530i	$0.110268\pi$
$7$			616.243i			0.679061i	−0.940595	+	0.339530i	$0.889732\pi$
$8$		0			0
$9$	460.056			0.210360
$10$		0			0
$11$			4625.43i			1.04780i	0.851780	+	0.523899i	$0.175523\pi$
$11$			4625.43i			1.04780i	−0.851780	+	0.523899i	$0.824477\pi$
$12$		0			0
$13$	1257.79	+	7820.90i	0.158783	+	0.987313i
$13$	1257.79	+	7820.90i	0.158783	+	0.987313i
$14$		0			0
$15$		−	25123.4i		−	1.92203i
$16$		0			0
$17$	−19536.2			−0.964427			−0.482214	−	0.876054i	$-0.660167\pi$
$17$	−19536.2			−0.964427			−0.482214	+	0.876054i	$0.660167\pi$
$18$		0			0
$19$			20077.3i			0.671533i	0.941945	+	0.335766i	$0.108995\pi$
$19$			20077.3i			0.671533i	−0.941945	+	0.335766i	$0.891005\pi$
$20$		0			0
$21$			31705.4i			0.747078i
$22$		0			0
$23$	−71368.5			−1.22309			−0.611546	−	0.791209i	$-0.709452\pi$
$23$	−71368.5			−1.22309			−0.611546	+	0.791209i	$0.709452\pi$
$24$		0			0
$25$	−160323.			−2.05214
$26$		0			0
$27$	−88850.5			−0.868734
$28$		0			0
$29$	138095.			1.05144			0.525722	−	0.850656i	$-0.323795\pi$
$29$	138095.			1.05144			0.525722	+	0.850656i	$0.323795\pi$
$30$		0			0
$31$		−	18285.7i		−	0.110242i	−0.998480	−	0.0551208i	$-0.982446\pi$
$31$		−	18285.7i		−	0.110242i	0.998480	−	0.0551208i	$-0.0175544\pi$
$32$		0			0
$33$			237976.i			1.15275i
$34$		0			0
$35$	300918.			1.18634
$36$		0			0
$37$			446035.i			1.44765i	0.689985	+	0.723824i	$0.257617\pi$
$37$			446035.i			1.44765i	−0.689985	+	0.723824i	$0.742383\pi$
$38$		0			0
$39$	64712.5	+	402382.i	0.174688	+	1.08621i
$40$		0			0
$41$		−	101639.i		−	0.230312i	−0.993347	−	0.115156i	$-0.963263\pi$
$41$		−	101639.i		−	0.230312i	0.993347	−	0.115156i	$-0.0367368\pi$
$42$		0			0
$43$	−103690.			−0.198883			−0.0994416	−	0.995043i	$-0.531706\pi$
$43$	−103690.			−0.198883			−0.0994416	+	0.995043i	$0.531706\pi$
$44$		0			0
$45$		−	224651.i		−	0.367506i
$46$		0			0
$47$			445129.i			0.625379i	0.949855	+	0.312690i	$0.101230\pi$
$47$			445129.i			0.625379i	−0.949855	+	0.312690i	$0.898770\pi$
$48$		0			0
$49$	443788.			0.538877
$50$		0			0
$51$	−1.00513e6			−1.06103
$52$		0			0
$53$	203578.			0.187830			0.0939149	−	0.995580i	$-0.470062\pi$
$53$	203578.			0.187830			0.0939149	+	0.995580i	$0.470062\pi$
$54$		0			0
$55$	2.25865e6			1.83054
$56$		0			0
$57$			1.03297e6i			0.738796i
$58$		0			0
$59$			2.50176e6i			1.58585i	0.609317	+	0.792927i	$0.291444\pi$
$59$			2.50176e6i			1.58585i	−0.609317	+	0.792927i	$0.708556\pi$
$60$		0			0
$61$	−792320.			−0.446937			−0.223468	−	0.974711i	$-0.571738\pi$
$61$	−792320.			−0.446937			−0.223468	+	0.974711i	$0.571738\pi$
$62$		0			0
$63$			283506.i			0.142847i
$64$		0			0
$65$	3.81904e6	−	614191.i	1.72487	−	0.277400i
$66$		0			0
$67$		−	3.17309e6i		−	1.28890i	−0.764645	−	0.644452i	$-0.777085\pi$
$67$		−	3.17309e6i		−	1.28890i	0.764645	−	0.644452i	$-0.222915\pi$
$68$		0			0
$69$	−3.67188e6			−1.34560
$70$		0			0
$71$			1.85875e6i			0.616333i	0.951332	+	0.308167i	$0.0997154\pi$
$71$			1.85875e6i			0.616333i	−0.951332	+	0.308167i	$0.900285\pi$
$72$		0			0
$73$		−	4.25734e6i		−	1.28088i	−0.768008	−	0.640440i	$-0.778752\pi$
$73$		−	4.25734e6i		−	1.28088i	0.768008	−	0.640440i	$-0.221248\pi$
$74$		0			0
$75$	−8.24856e6			−2.25769
$76$		0			0
$77$	−2.85039e6			−0.711519
$78$		0			0
$79$	−2.38970e6			−0.545317			−0.272658	−	0.962111i	$-0.587903\pi$
$79$	−2.38970e6			−0.545317			−0.272658	+	0.962111i	$0.587903\pi$
$80$		0			0
$81$	−5.57746e6			−1.16611
$82$		0			0
$83$		−	1.12536e6i		−	0.216032i	−0.994149	−	0.108016i	$-0.965550\pi$
$83$		−	1.12536e6i		−	0.216032i	0.994149	−	0.108016i	$-0.0344498\pi$
$84$		0			0
$85$			9.53976e6i			1.68489i
$86$		0			0
$87$	7.10495e6			1.15676
$88$		0			0
$89$			8.00227e6i			1.20323i	0.798787	+	0.601614i	$0.205476\pi$
$89$			8.00227e6i			1.20323i	−0.798787	+	0.601614i	$0.794524\pi$
$90$		0			0
$91$	−4.81957e6	+	775101.i	−0.670446	+	0.107823i
$92$		0			0
$93$		−	940791.i		−	0.121284i
$94$		0			0
$95$	9.80397e6			1.17319
$96$		0			0
$97$			276982.i			0.0308141i	0.999881	+	0.0154071i	$0.00490442\pi$
$97$			276982.i			0.0308141i	−0.999881	+	0.0154071i	$0.995096\pi$
$98$		0			0
$99$			2.12796e6i			0.220414i
$100$		0			0
$101$	1.32796e7			1.28251			0.641256	−	0.767327i	$-0.278414\pi$
$101$	1.32796e7			1.28251			0.641256	+	0.767327i	$0.278414\pi$
$102$		0			0
$103$	7.07119e6			0.637621			0.318810	−	0.947819i	$-0.396717\pi$
$103$	7.07119e6			0.637621			0.318810	+	0.947819i	$0.396717\pi$
$104$		0			0
$105$	1.54821e7			1.30517
$106$		0			0
$107$	−2.81264e6			−0.221958			−0.110979	−	0.993823i	$-0.535399\pi$
$107$	−2.81264e6			−0.221958			−0.110979	+	0.993823i	$0.535399\pi$
$108$		0			0
$109$			1.77688e7i			1.31421i	0.753798	+	0.657107i	$0.228220\pi$
$109$			1.77688e7i			1.31421i	−0.753798	+	0.657107i	$0.771780\pi$
$110$		0			0
$111$			2.29483e7i			1.59265i
$112$		0			0
$113$	−1.79731e7			−1.17179			−0.585894	−	0.810388i	$-0.699257\pi$
$113$	−1.79731e7			−1.17179			−0.585894	+	0.810388i	$0.699257\pi$
$114$		0			0
$115$			3.48501e7i			2.13679i
$116$		0			0
$117$	578652.	+	3.59805e6i	0.0334016	+	0.207691i
$118$		0			0
$119$		−	1.20391e7i		−	0.654904i
$120$		0			0
$121$	−1.90744e6			−0.0978819
$122$		0			0
$123$		−	5.22928e6i		−	0.253381i
$124$		0			0
$125$			4.01383e7i			1.83812i
$126$		0			0
$127$	−2.01354e7			−0.872263			−0.436131	−	0.899883i	$-0.643652\pi$
$127$	−2.01354e7			−0.872263			−0.436131	+	0.899883i	$0.643652\pi$
$128$		0			0
$129$	−5.33482e6			−0.218804
$130$		0			0
$131$	2.41394e7			0.938159			0.469079	−	0.883156i	$-0.344586\pi$
$131$	2.41394e7			0.938159			0.469079	+	0.883156i	$0.344586\pi$
$132$		0			0
$133$	−1.23725e7			−0.456011
$134$		0			0
$135$			4.33867e7i			1.51771i
$136$		0			0
$137$			8.35854e6i			0.277721i	0.990312	+	0.138860i	$0.0443439\pi$
$137$			8.35854e6i			0.277721i	−0.990312	+	0.138860i	$0.955656\pi$
$138$		0			0
$139$	−4.59389e7			−1.45087			−0.725435	−	0.688290i	$-0.758361\pi$
$139$	−4.59389e7			−1.45087			−0.725435	+	0.688290i	$0.758361\pi$
$140$		0			0
$141$			2.29017e7i			0.688020i
$142$		0			0
$143$	−3.61750e7	+	5.81780e6i	−1.03451	+	0.166373i
$144$		0			0
$145$		−	6.74336e7i		−	1.83691i
$146$		0			0
$147$	2.28327e7			0.592852
$148$		0			0
$149$			7.52366e6i			0.186328i	0.995651	+	0.0931638i	$0.0296980\pi$
$149$			7.52366e6i			0.186328i	−0.995651	+	0.0931638i	$0.970302\pi$
$150$		0			0
$151$			5.79965e7i			1.37083i	0.728155	+	0.685413i	$0.240378\pi$
$151$			5.79965e7i			1.37083i	−0.728155	+	0.685413i	$0.759622\pi$
$152$		0			0
$153$	−8.98776e6			−0.202876
$154$		0			0
$155$	−8.92912e6			−0.192596
$156$		0			0
$157$	−8.29958e7			−1.71162			−0.855810	−	0.517291i	$-0.826941\pi$
$157$	−8.29958e7			−1.71162			−0.855810	+	0.517291i	$0.826941\pi$
$158$		0			0
$159$	1.04740e7			0.206643
$160$		0			0
$161$		−	4.39803e7i		−	0.830553i
$162$		0			0
$163$		−	3.36561e7i		−	0.608705i	−0.952560	−	0.304352i	$-0.901560\pi$
$163$		−	3.36561e7i		−	0.608705i	0.952560	−	0.304352i	$-0.0984400\pi$
$164$		0			0
$165$	1.16207e8			2.01390
$166$		0			0
$167$		−	7.76344e7i		−	1.28987i	−0.764237	−	0.644936i	$-0.776884\pi$
$167$		−	7.76344e7i		−	1.28987i	0.764237	−	0.644936i	$-0.223116\pi$
$168$		0			0
$169$	−5.95845e7	+	1.96740e7i	−0.949576	+	0.313538i
$170$		0			0
$171$			9.23668e6i			0.141263i
$172$		0			0
$173$	−668606.			−0.00981768			−0.00490884	−	0.999988i	$-0.501563\pi$
$173$	−668606.			−0.00981768			−0.00490884	+	0.999988i	$0.501563\pi$
$174$		0			0
$175$		−	9.87980e7i		−	1.39353i
$176$		0			0
$177$			1.28714e8i			1.74470i
$178$		0			0
$179$	1.05607e8			1.37628			0.688142	−	0.725576i	$-0.258427\pi$
$179$	1.05607e8			1.37628			0.688142	+	0.725576i	$0.258427\pi$
$180$		0			0
$181$	−1.42306e7			−0.178381			−0.0891904	−	0.996015i	$-0.528428\pi$
$181$	−1.42306e7			−0.178381			−0.0891904	+	0.996015i	$0.528428\pi$
$182$		0			0
$183$	−4.07645e7			−0.491703
$184$		0			0
$185$	2.17804e8			2.52909
$186$		0			0
$187$		−	9.03635e7i		−	1.01053i
$188$		0			0
$189$		−	5.47534e7i		−	0.589923i
$190$		0			0
$191$	6.06460e7			0.629775			0.314887	−	0.949129i	$-0.398033\pi$
$191$	6.06460e7			0.629775			0.314887	+	0.949129i	$0.398033\pi$
$192$		0			0
$193$		−	1.96802e8i		−	1.97051i	−0.171102	−	0.985253i	$-0.554733\pi$
$193$		−	1.96802e8i		−	1.97051i	0.171102	−	0.985253i	$-0.445267\pi$
$194$		0			0
$195$	1.96488e8	−	3.15999e7i	1.89764	−	0.305186i
$196$		0			0
$197$			7.08567e7i			0.660312i	0.943926	+	0.330156i	$0.107101\pi$
$197$			7.08567e7i			0.660312i	−0.943926	+	0.330156i	$0.892899\pi$
$198$		0			0
$199$	−1.26370e8			−1.13674			−0.568368	−	0.822774i	$-0.692425\pi$
$199$	−1.26370e8			−1.13674			−0.568368	+	0.822774i	$0.692425\pi$
$200$		0			0
$201$		−	1.63254e8i		−	1.41801i
$202$		0			0
$203$			8.51003e7i			0.713995i
$204$		0			0
$205$	−4.96315e7			−0.402364
$206$		0			0
$207$	−3.28335e7			−0.257289
$208$		0			0
$209$	−9.28660e7			−0.703631
$210$		0			0
$211$	−1.08405e8			−0.794440			−0.397220	−	0.917723i	$-0.630025\pi$
$211$	−1.08405e8			−0.794440			−0.397220	+	0.917723i	$0.630025\pi$
$212$		0			0
$213$			9.56316e7i			0.678067i
$214$		0			0
$215$			5.06332e7i			0.347456i
$216$		0			0
$217$	1.12684e7			0.0748607
$218$		0			0
$219$		−	2.19038e8i		−	1.40918i
$220$		0			0
$221$	−2.45724e7	−	1.52791e8i	−0.153135	−	0.952192i
$222$		0			0
$223$			4.34061e7i			0.262110i	0.991375	+	0.131055i	$0.0418364\pi$
$223$			4.34061e7i			0.262110i	−0.991375	+	0.131055i	$0.958164\pi$
$224$		0			0
$225$	−7.37577e7			−0.431687
$226$		0			0
$227$		−	1.30814e8i		−	0.742272i	−0.928579	−	0.371136i	$-0.878968\pi$
$227$		−	1.30814e8i		−	0.742272i	0.928579	−	0.371136i	$-0.121032\pi$
$228$		0			0
$229$		−	1.82131e7i		−	0.100221i	−0.998744	−	0.0501105i	$-0.984043\pi$
$229$		−	1.82131e7i		−	0.100221i	0.998744	−	0.0501105i	$-0.0159573\pi$
$230$		0			0
$231$	−1.46651e8			−0.782787
$232$		0			0
$233$	1.76958e8			0.916482			0.458241	−	0.888828i	$-0.348480\pi$
$233$	1.76958e8			0.916482			0.458241	+	0.888828i	$0.348480\pi$
$234$		0			0
$235$	2.17362e8			1.09256
$236$		0			0
$237$	−1.22949e8			−0.599937
$238$		0			0
$239$		−	7.32549e7i		−	0.347092i	−0.984826	−	0.173546i	$-0.944478\pi$
$239$		−	7.32549e7i		−	0.347092i	0.984826	−	0.173546i	$-0.0555225\pi$
$240$		0			0
$241$			2.20158e8i			1.01315i	0.862195	+	0.506576i	$0.169089\pi$
$241$			2.20158e8i			1.01315i	−0.862195	+	0.506576i	$0.830911\pi$
$242$		0			0
$243$	−9.26418e7			−0.414176
$244$		0			0
$245$		−	2.16707e8i		−	0.941437i
$246$		0			0
$247$	−1.57022e8	+	2.52529e7i	−0.663013	+	0.106628i
$248$		0			0
$249$		−	5.78992e7i		−	0.237670i
$250$		0			0
$251$	1.29959e8			0.518738			0.259369	−	0.965778i	$-0.416485\pi$
$251$	1.29959e8			0.518738			0.259369	+	0.965778i	$0.416485\pi$
$252$		0			0
$253$		−	3.30110e8i		−	1.28155i
$254$		0			0
$255$			4.90817e8i			1.85365i
$256$		0			0
$257$	1.11340e6			0.00409153			0.00204577	−	0.999998i	$-0.499349\pi$
$257$	1.11340e6			0.00409153			0.00204577	+	0.999998i	$0.499349\pi$
$258$		0			0
$259$	−2.74866e8			−0.983040
$260$		0			0
$261$	6.35317e7			0.221181
$262$		0			0
$263$	3.02037e8			1.02380			0.511900	−	0.859045i	$-0.328942\pi$
$263$	3.02037e8			1.02380			0.511900	+	0.859045i	$0.328942\pi$
$264$		0			0
$265$		−	9.94093e7i		−	0.328145i
$266$		0			0
$267$			4.11713e8i			1.32375i
$268$		0			0
$269$	−2.42963e8			−0.761040			−0.380520	−	0.924773i	$-0.624255\pi$
$269$	−2.42963e8			−0.761040			−0.380520	+	0.924773i	$0.624255\pi$
$270$		0			0
$271$		−	3.31493e8i		−	1.01177i	−0.862601	−	0.505886i	$-0.831166\pi$
$271$		−	3.31493e8i		−	1.01177i	0.862601	−	0.505886i	$-0.168834\pi$
$272$		0			0
$273$	−2.47965e8	+	3.98786e7i	−0.737600	+	0.118623i
$274$		0			0
$275$		−	7.41564e8i		−	2.15023i
$276$		0			0
$277$	2.92754e8			0.827607			0.413803	−	0.910366i	$-0.364200\pi$
$277$	2.92754e8			0.827607			0.413803	+	0.910366i	$0.364200\pi$
$278$		0			0
$279$		−	8.41245e6i		−	0.0231904i
$280$		0			0
$281$			4.81806e8i			1.29539i	0.761901	+	0.647694i	$0.224266\pi$
$281$			4.81806e8i			1.29539i	−0.761901	+	0.647694i	$0.775734\pi$
$282$		0			0
$283$	−6.21029e8			−1.62877			−0.814385	−	0.580326i	$-0.802925\pi$
$283$	−6.21029e8			−1.62877			−0.814385	+	0.580326i	$0.802925\pi$
$284$		0			0
$285$	5.04410e8			1.29070
$286$		0			0
$287$	6.26343e7			0.156396
$288$		0			0
$289$	−2.86746e7			−0.0698804
$290$		0			0
$291$			1.42506e7i			0.0339006i
$292$		0			0
$293$			2.25288e8i			0.523240i	0.965171	+	0.261620i	$0.0842568\pi$
$293$			2.25288e8i			0.523240i	−0.965171	+	0.261620i	$0.915743\pi$
$294$		0			0
$295$	1.22164e9			2.77054
$296$		0			0
$297$		−	4.10972e8i		−	0.910258i
$298$		0			0
$299$	−8.97662e7	−	5.58166e8i	−0.194206	−	1.20757i
$300$		0			0
$301$		−	6.38983e7i		−	0.135054i
$302$		0			0
$303$	6.83232e8			1.41097
$304$		0			0
$305$			3.86899e8i			0.780815i
$306$		0			0
$307$		−	1.94968e8i		−	0.384572i	−0.981339	−	0.192286i	$-0.938410\pi$
$307$		−	1.94968e8i		−	0.384572i	0.981339	−	0.192286i	$-0.0615902\pi$
$308$		0			0
$309$	3.63810e8			0.701487
$310$		0			0
$311$	−3.23645e8			−0.610110			−0.305055	−	0.952335i	$-0.598675\pi$
$311$	−3.23645e8			−0.610110			−0.305055	+	0.952335i	$0.598675\pi$
$312$		0			0
$313$	1.52942e8			0.281917			0.140959	−	0.990016i	$-0.454982\pi$
$313$	1.52942e8			0.281917			0.140959	+	0.990016i	$0.454982\pi$
$314$		0			0
$315$	1.38439e8			0.249559
$316$		0			0
$317$			1.05459e8i			0.185941i	0.995669	+	0.0929707i	$0.0296363\pi$
$317$			1.05459e8i			0.185941i	−0.995669	+	0.0929707i	$0.970364\pi$
$318$		0			0
$319$			6.38751e8i			1.10170i
$320$		0			0
$321$	−1.44709e8			−0.244190
$322$		0			0
$323$		−	3.92234e8i		−	0.647644i
$324$		0			0
$325$	−2.01652e8	−	1.25387e9i	−0.325845	−	2.02610i
$326$		0			0
$327$			9.14198e8i			1.44585i
$328$		0			0
$329$	−2.74308e8			−0.424670
$330$		0			0
$331$		−	6.53937e8i		−	0.991146i	−0.868566	−	0.495573i	$-0.834958\pi$
$331$		−	6.53937e8i		−	0.991146i	0.868566	−	0.495573i	$-0.165042\pi$
$332$		0			0
$333$			2.05201e8i			0.304526i
$334$		0			0
$335$	−1.54946e9			−2.25176
$336$		0			0
$337$	−6.58782e8			−0.937642			−0.468821	−	0.883293i	$-0.655321\pi$
$337$	−6.58782e8			−0.937642			−0.468821	+	0.883293i	$0.655321\pi$
$338$		0			0
$339$	−9.24709e8			−1.28916
$340$		0			0
$341$	8.45792e7			0.115511
$342$		0			0
$343$			7.80983e8i			1.04499i
$344$		0			0
$345$			1.79302e9i			2.35081i
$346$		0			0
$347$	3.76864e8			0.484208			0.242104	−	0.970250i	$-0.422163\pi$
$347$	3.76864e8			0.484208			0.242104	+	0.970250i	$0.422163\pi$
$348$		0			0
$349$			7.32436e8i			0.922318i	0.887317	+	0.461159i	$0.152566\pi$
$349$			7.32436e8i			0.922318i	−0.887317	+	0.461159i	$0.847434\pi$
$350$		0			0
$351$	−1.11755e8	−	6.94891e8i	−0.137940	−	0.857712i
$352$		0			0
$353$		−	1.08614e9i		−	1.31424i	−0.753788	−	0.657118i	$-0.771775\pi$
$353$		−	1.08614e9i		−	1.31424i	0.753788	−	0.657118i	$-0.228225\pi$
$354$		0			0
$355$	9.07647e8			1.07676
$356$		0			0
$357$		−	6.19404e8i		−	0.720502i
$358$		0			0
$359$			6.86701e8i			0.783316i	0.920111	+	0.391658i	$0.128098\pi$
$359$			6.86701e8i			0.783316i	−0.920111	+	0.391658i	$0.871902\pi$
$360$		0			0
$361$	4.90775e8			0.549044
$362$		0			0
$363$	−9.81370e7			−0.107686
$364$		0			0
$365$	−2.07891e9			−2.23775
$366$		0			0
$367$	−4.74836e8			−0.501432			−0.250716	−	0.968061i	$-0.580666\pi$
$367$	−4.74836e8			−0.501432			−0.250716	+	0.968061i	$0.580666\pi$
$368$		0			0
$369$		−	4.67597e7i		−	0.0484484i
$370$		0			0
$371$			1.25453e8i			0.127548i
$372$		0			0
$373$	1.55637e9			1.55286			0.776431	−	0.630202i	$-0.217028\pi$
$373$	1.55637e9			1.55286			0.776431	+	0.630202i	$0.217028\pi$
$374$		0			0
$375$			2.06510e9i			2.02224i
$376$		0			0
$377$	1.73694e8	+	1.08003e9i	0.166952	+	1.03811i
$378$		0			0
$379$			1.39184e8i			0.131326i	0.997842	+	0.0656631i	$0.0209163\pi$
$379$			1.39184e8i			0.131326i	−0.997842	+	0.0656631i	$0.979084\pi$
$380$		0			0
$381$	−1.03596e9			−0.959631
$382$		0			0
$383$			9.91073e8i			0.901385i	0.892679	+	0.450692i	$0.148823\pi$
$383$			9.91073e8i			0.901385i	−0.892679	+	0.450692i	$0.851177\pi$
$384$		0			0
$385$			1.39188e9i			1.24305i
$386$		0			0
$387$	−4.77034e7			−0.0418370
$388$		0			0
$389$	9.01206e8			0.776248			0.388124	−	0.921607i	$-0.373123\pi$
$389$	9.01206e8			0.776248			0.388124	+	0.921607i	$0.373123\pi$
$390$		0			0
$391$	1.39427e9			1.17958
$392$		0			0
$393$	1.24196e9			1.03213
$394$		0			0
$395$			1.16692e9i			0.952688i
$396$		0			0
$397$		−	9.85599e8i		−	0.790558i	−0.918561	−	0.395279i	$-0.870648\pi$
$397$		−	9.85599e8i		−	0.790558i	0.918561	−	0.395279i	$-0.129352\pi$
$398$		0			0
$399$	−6.36558e8			−0.501687
$400$		0			0
$401$		−	2.69423e8i		−	0.208655i	−0.994543	−	0.104328i	$-0.966731\pi$
$401$		−	2.69423e8i		−	0.208655i	0.994543	−	0.104328i	$-0.0332690\pi$
$402$		0			0
$403$	1.43011e8	−	2.29995e7i	0.108843	−	0.0175045i
$404$		0			0
$405$			2.72354e9i			2.03723i
$406$		0			0
$407$	−2.06310e9			−1.51684
$408$		0			0
$409$		−	3.40992e8i		−	0.246441i	−0.992379	−	0.123220i	$-0.960678\pi$
$409$		−	3.40992e8i		−	0.246441i	0.992379	−	0.123220i	$-0.0393222\pi$
$410$		0			0
$411$			4.30043e8i			0.305538i
$412$		0			0
$413$	−1.54169e9			−1.07689
$414$		0			0
$415$	−5.49526e8			−0.377416
$416$		0			0
$417$	−2.36353e9			−1.59619
$418$		0			0
$419$	2.32463e9			1.54385			0.771923	−	0.635716i	$-0.219295\pi$
$419$	2.32463e9			1.54385			0.771923	+	0.635716i	$0.219295\pi$
$420$		0			0
$421$			2.46436e9i			1.60959i	0.593551	+	0.804797i	$0.297726\pi$
$421$			2.46436e9i			1.60959i	−0.593551	+	0.804797i	$0.702274\pi$
$422$		0			0
$423$			2.04784e8i			0.131555i
$424$		0			0
$425$	3.13211e9			1.97914
$426$		0			0
$427$		−	4.88261e8i		−	0.303497i
$428$		0			0
$429$	−1.86119e9	+	2.99323e8i	−1.13813	+	0.183037i
$430$		0			0
$431$			8.64638e8i			0.520192i	0.965583	+	0.260096i	$0.0837543\pi$
$431$			8.64638e8i			0.520192i	−0.965583	+	0.260096i	$0.916246\pi$
$432$		0			0
$433$	−3.99758e8			−0.236641			−0.118320	−	0.992975i	$-0.537751\pi$
$433$	−3.99758e8			−0.236641			−0.118320	+	0.992975i	$0.537751\pi$
$434$		0			0
$435$		−	3.46943e9i		−	2.02090i
$436$		0			0
$437$		−	1.43288e9i		−	0.821346i
$438$		0			0
$439$	1.30846e8			0.0738130			0.0369065	−	0.999319i	$-0.488250\pi$
$439$	1.30846e8			0.0738130			0.0369065	+	0.999319i	$0.488250\pi$
$440$		0			0
$441$	2.04168e8			0.113358
$442$		0			0
$443$	6.35046e8			0.347050			0.173525	−	0.984829i	$-0.444484\pi$
$443$	6.35046e8			0.347050			0.173525	+	0.984829i	$0.444484\pi$
$444$		0			0
$445$	3.90760e9			2.10208
$446$		0			0
$447$			3.87089e8i			0.204991i
$448$		0			0
$449$		−	6.68424e8i		−	0.348490i	−0.984702	−	0.174245i	$-0.944252\pi$
$449$		−	6.68424e8i		−	0.348490i	0.984702	−	0.174245i	$-0.0557484\pi$
$450$		0			0
$451$	4.70124e8			0.241321
$452$		0			0
$453$			2.98389e9i			1.50813i
$454$		0			0
$455$	3.78491e8	+	2.35345e9i	0.188372	+	1.17129i
$456$		0			0
$457$		−	1.09854e9i		−	0.538406i	−0.963083	−	0.269203i	$-0.913240\pi$
$457$		−	1.09854e9i		−	0.538406i	0.963083	−	0.269203i	$-0.0867602\pi$
$458$		0			0
$459$	1.73580e9			0.837830
$460$		0			0
$461$		−	1.96364e9i		−	0.933489i	−0.884392	−	0.466745i	$-0.845427\pi$
$461$		−	1.96364e9i		−	0.933489i	0.884392	−	0.466745i	$-0.154573\pi$
$462$		0			0
$463$			4.36528e6i			0.00204399i	0.999999	+	0.00102199i	$0.000325311\pi$
$463$			4.36528e6i			0.00204399i	−0.999999	+	0.00102199i	$0.999675\pi$
$464$		0			0
$465$	−4.59399e8			−0.211887
$466$		0			0
$467$	1.77366e9			0.805861			0.402931	−	0.915230i	$-0.367992\pi$
$467$	1.77366e9			0.805861			0.402931	+	0.915230i	$0.367992\pi$
$468$		0			0
$469$	1.95539e9			0.875244
$470$		0			0
$471$	−4.27009e9			−1.88306
$472$		0			0
$473$		−	4.79612e8i		−	0.208390i
$474$		0			0
$475$		−	3.21885e9i		−	1.37808i
$476$		0			0
$477$	9.36571e7			0.0395118
$478$		0			0
$479$		−	2.31639e9i		−	0.963026i	−0.876439	−	0.481513i	$-0.840087\pi$
$479$		−	2.31639e9i		−	0.963026i	0.876439	−	0.481513i	$-0.159913\pi$
$480$		0			0
$481$	−3.48839e9	+	5.61016e8i	−1.42928	+	0.229862i
$482$		0			0
$483$		−	2.26277e9i		−	0.913744i
$484$		0			0
$485$	1.35253e8			0.0538334
$486$		0			0
$487$			2.04846e9i			0.803668i	0.915713	+	0.401834i	$0.131627\pi$
$487$			2.04846e9i			0.803668i	−0.915713	+	0.401834i	$0.868373\pi$
$488$		0			0
$489$		−	1.73159e9i		−	0.669675i
$490$		0			0
$491$	−3.64385e9			−1.38923			−0.694616	−	0.719381i	$-0.744426\pi$
$491$	−3.64385e9			−1.38923			−0.694616	+	0.719381i	$0.744426\pi$
$492$		0			0
$493$	−2.69786e9			−1.01404
$494$		0			0
$495$	1.03911e9			0.385072
$496$		0			0
$497$	−1.14544e9			−0.418528
$498$		0			0
$499$		−	4.34184e9i		−	1.56431i	−0.623086	−	0.782153i	$-0.714121\pi$
$499$		−	4.34184e9i		−	1.56431i	0.623086	−	0.782153i	$-0.285879\pi$
$500$		0			0
$501$		−	3.99425e9i		−	1.41907i
$502$		0			0
$503$	−1.29100e9			−0.452311			−0.226155	−	0.974091i	$-0.572616\pi$
$503$	−1.29100e9			−0.452311			−0.226155	+	0.974091i	$0.572616\pi$
$504$		0			0
$505$		−	6.48460e9i		−	2.24060i
$506$		0			0
$507$	−3.06559e9	+	1.01222e9i	−1.04469	+	0.344943i
$508$		0			0
$509$			4.23593e9i			1.42376i	0.702301	+	0.711880i	$0.252156\pi$
$509$			4.23593e9i			1.42376i	−0.702301	+	0.711880i	$0.747844\pi$
$510$		0			0
$511$	2.62356e9			0.869795
$512$		0			0
$513$		−	1.78388e9i		−	0.583383i
$514$		0			0
$515$		−	3.45294e9i		−	1.11395i
$516$		0			0
$517$	−2.05891e9			−0.655272
$518$		0			0
$519$	−3.43995e7			−0.0108011
$520$		0			0
$521$	−2.70853e9			−0.839076			−0.419538	−	0.907738i	$-0.637808\pi$
$521$	−2.70853e9			−0.839076			−0.419538	+	0.907738i	$0.637808\pi$
$522$		0			0
$523$	2.07895e9			0.635461			0.317731	−	0.948181i	$-0.397079\pi$
$523$	2.07895e9			0.635461			0.317731	+	0.948181i	$0.397079\pi$
$524$		0			0
$525$		−	5.08311e9i		−	1.53311i
$526$		0			0
$527$			3.57233e8i			0.106320i
$528$		0			0
$529$	1.68863e9			0.495953
$530$		0			0
$531$			1.15095e9i			0.333599i
$532$		0			0
$533$	7.94909e8	−	1.27840e8i	0.227390	−	0.0365697i
$534$		0			0
$535$			1.37345e9i			0.387769i
$536$		0			0
$537$	5.43344e9			1.51414
$538$		0			0
$539$			2.05271e9i			0.564634i
$540$		0			0
$541$		−	3.70324e9i		−	1.00552i	−0.864426	−	0.502761i	$-0.832318\pi$
$541$		−	3.70324e9i		−	1.00552i	0.864426	−	0.502761i	$-0.167682\pi$
$542$		0			0
$543$	−7.32158e8			−0.196248
$544$		0			0
$545$	8.67672e9			2.29598
$546$		0			0
$547$	−2.64179e9			−0.690149			−0.345074	−	0.938575i	$-0.612146\pi$
$547$	−2.64179e9			−0.690149			−0.345074	+	0.938575i	$0.612146\pi$
$548$		0			0
$549$	−3.64512e8			−0.0940174
$550$		0			0
$551$			2.77258e9i			0.706080i
$552$		0			0
$553$		−	1.47264e9i		−	0.370303i
$554$		0			0
$555$	1.12059e10			2.78242
$556$		0			0
$557$			5.09926e9i			1.25030i	0.780505	+	0.625149i	$0.214962\pi$
$557$			5.09926e9i			1.25030i	−0.780505	+	0.625149i	$0.785038\pi$
$558$		0			0
$559$	−1.30420e8	−	8.10951e8i	−0.0315793	−	0.196360i
$560$		0			0
$561$		−	4.64916e9i		−	1.11174i
$562$		0			0
$563$	1.20149e8			0.0283754			0.0141877	−	0.999899i	$-0.495484\pi$
$563$	1.20149e8			0.0283754			0.0141877	+	0.999899i	$0.495484\pi$
$564$		0			0
$565$			8.77649e9i			2.04716i
$566$		0			0
$567$		−	3.43707e9i		−	0.791858i
$568$		0			0
$569$	−4.82795e9			−1.09868			−0.549338	−	0.835600i	$-0.685120\pi$
$569$	−4.82795e9			−1.09868			−0.549338	+	0.835600i	$0.685120\pi$
$570$		0			0
$571$	−6.31138e9			−1.41872			−0.709362	−	0.704845i	$-0.751017\pi$
$571$	−6.31138e9			−1.41872			−0.709362	+	0.704845i	$0.751017\pi$
$572$		0			0
$573$	3.12021e9			0.692855
$574$		0			0
$575$	1.14420e10			2.50995
$576$		0			0
$577$			6.12272e9i			1.32687i	0.748233	+	0.663436i	$0.230903\pi$
$577$			6.12272e9i			1.32687i	−0.748233	+	0.663436i	$0.769097\pi$
$578$		0			0
$579$		−	1.01254e10i		−	2.16788i
$580$		0			0
$581$	6.93494e8			0.146699
$582$		0			0
$583$			9.41634e8i			0.196808i
$584$		0			0
$585$	1.75697e9	−	2.82562e8i	0.362843	−	0.0583538i
$586$		0			0
$587$			1.48927e9i			0.303907i	0.988388	+	0.151953i	$0.0485563\pi$
$587$			1.48927e9i			0.303907i	−0.988388	+	0.151953i	$0.951444\pi$
$588$		0			0
$589$	3.67127e8			0.0740309
$590$		0			0
$591$			3.64555e9i			0.726451i
$592$		0			0
$593$		−	5.49424e9i		−	1.08197i	−0.841031	−	0.540987i	$-0.818051\pi$
$593$		−	5.49424e9i		−	1.08197i	0.841031	−	0.540987i	$-0.181949\pi$
$594$		0			0
$595$	−5.87881e9			−1.14414
$596$		0			0
$597$	−6.50170e9			−1.25060
$598$		0			0
$599$	−4.52708e9			−0.860645			−0.430322	−	0.902675i	$-0.641600\pi$
$599$	−4.52708e9			−0.860645			−0.430322	+	0.902675i	$0.641600\pi$
$600$		0			0
$601$	−2.93232e9			−0.550998			−0.275499	−	0.961301i	$-0.588843\pi$
$601$	−2.93232e9			−0.550998			−0.275499	+	0.961301i	$0.588843\pi$
$602$		0			0
$603$		−	1.45980e9i		−	0.271133i
$604$		0			0
$605$			9.31425e8i			0.171003i
$606$		0			0
$607$	1.02999e10			1.86927			0.934637	−	0.355604i	$-0.115725\pi$
$607$	1.02999e10			1.86927			0.934637	+	0.355604i	$0.115725\pi$
$608$		0			0
$609$			4.37837e9i			0.785511i
$610$		0			0
$611$	−3.48131e9	+	5.59877e8i	−0.617446	+	0.0992998i
$612$		0			0
$613$		−	7.93265e9i		−	1.39093i	−0.718558	−	0.695467i	$-0.755197\pi$
$613$		−	7.93265e9i		−	1.39093i	0.718558	−	0.695467i	$-0.244803\pi$
$614$		0			0
$615$	−2.55352e9			−0.442666
$616$		0			0
$617$			8.87840e9i			1.52173i	0.648912	+	0.760863i	$0.275224\pi$
$617$			8.87840e9i			1.52173i	−0.648912	+	0.760863i	$0.724776\pi$
$618$		0			0
$619$		−	7.70576e9i		−	1.30586i	−0.757416	−	0.652932i	$-0.773539\pi$
$619$		−	7.70576e9i		−	1.30586i	0.757416	−	0.652932i	$-0.226461\pi$
$620$		0			0
$621$	6.34112e9			1.06254
$622$		0			0
$623$	−4.93134e9			−0.817065
$624$		0			0
$625$	7.07477e9			1.15913
$626$		0			0
$627$	−4.77792e9			−0.774109
$628$		0			0
$629$		−	8.71384e9i		−	1.39615i
$630$		0			0
$631$			1.12137e10i			1.77683i	0.459045	+	0.888413i	$0.348192\pi$
$631$			1.12137e10i			1.77683i	−0.459045	+	0.888413i	$0.651808\pi$
$632$		0			0
$633$	−5.57739e9			−0.874014
$634$		0			0
$635$			9.83235e9i			1.52387i
$636$		0			0
$637$	5.58190e8	+	3.47082e9i	0.0855646	+	0.532040i
$638$		0			0
$639$			8.55127e8i			0.129652i
$640$		0			0
$641$	−9.37779e9			−1.40636			−0.703181	−	0.711011i	$-0.748238\pi$
$641$	−9.37779e9			−1.40636			−0.703181	+	0.711011i	$0.748238\pi$
$642$		0			0
$643$		−	6.09443e8i		−	0.0904054i	−0.998978	−	0.0452027i	$-0.985607\pi$
$643$		−	6.09443e8i		−	0.0904054i	0.998978	−	0.0452027i	$-0.0143934\pi$
$644$		0			0
$645$			2.60505e9i			0.382259i
$646$		0			0
$647$	3.63441e9			0.527556			0.263778	−	0.964583i	$-0.415031\pi$
$647$	3.63441e9			0.527556			0.263778	+	0.964583i	$0.415031\pi$
$648$		0			0
$649$	−1.15717e10			−1.66166
$650$		0			0
$651$	5.79755e8			0.0823591
$652$		0			0
$653$	−3.71780e9			−0.522505			−0.261252	−	0.965271i	$-0.584135\pi$
$653$	−3.71780e9			−0.522505			−0.261252	+	0.965271i	$0.584135\pi$
$654$		0			0
$655$		−	1.17875e10i		−	1.63900i
$656$		0			0
$657$		−	1.95862e9i		−	0.269445i
$658$		0			0
$659$	−6.01769e9			−0.819088			−0.409544	−	0.912290i	$-0.634312\pi$
$659$	−6.01769e9			−0.819088			−0.409544	+	0.912290i	$0.634312\pi$
$660$		0			0
$661$		−	1.36622e9i		−	0.183999i	−0.995759	−	0.0919993i	$-0.970674\pi$
$661$		−	1.36622e9i		−	0.183999i	0.995759	−	0.0919993i	$-0.0293257\pi$
$662$		0			0
$663$	−1.26424e9	−	7.86102e9i	−0.168473	−	1.04757i
$664$		0			0
$665$			6.04162e9i			0.796669i
$666$		0			0
$667$	−9.85566e9			−1.28601
$668$		0			0
$669$			2.23322e9i			0.288364i
$670$		0			0
$671$		−	3.66482e9i		−	0.468300i
$672$		0			0
$673$	1.13200e9			0.143150			0.0715751	−	0.997435i	$-0.477197\pi$
$673$	1.13200e9			0.143150			0.0715751	+	0.997435i	$0.477197\pi$
$674$		0			0
$675$	1.42448e10			1.78276
$676$		0			0
$677$	2.50487e9			0.310259			0.155130	−	0.987894i	$-0.450420\pi$
$677$	2.50487e9			0.310259			0.155130	+	0.987894i	$0.450420\pi$
$678$		0			0
$679$	−1.70688e8			−0.0209247
$680$		0			0
$681$		−	6.73031e9i		−	0.816620i
$682$		0			0
$683$			1.16246e9i			0.139606i	0.997561	+	0.0698032i	$0.0222371\pi$
$683$			1.16246e9i			0.139606i	−0.997561	+	0.0698032i	$0.977763\pi$
$684$		0			0
$685$	4.08157e9			0.485188
$686$		0			0
$687$		−	9.37053e8i		−	0.110259i
$688$		0			0
$689$	2.56057e8	+	1.59216e9i	0.0298242	+	0.185447i
$690$		0			0
$691$		−	1.00279e10i		−	1.15621i	−0.815963	−	0.578104i	$-0.803793\pi$
$691$		−	1.00279e10i		−	1.15621i	0.815963	−	0.578104i	$-0.196207\pi$
$692$		0			0
$693$	−1.31134e9			−0.149675
$694$		0			0
$695$			2.24325e10i			2.53472i
$696$		0			0
$697$			1.98564e9i			0.222119i
$698$		0			0
$699$	9.10440e9			1.00828
$700$		0			0
$701$	1.58959e10			1.74290			0.871449	−	0.490486i	$-0.163181\pi$
$701$	1.58959e10			1.74290			0.871449	+	0.490486i	$0.163181\pi$
$702$		0			0
$703$	−8.95516e9			−0.972142
$704$		0			0
$705$	1.11832e10			1.20200
$706$		0			0
$707$			8.18348e9i			0.870904i
$708$		0			0
$709$		−	3.06676e9i		−	0.323160i	−0.986860	−	0.161580i	$-0.948341\pi$
$709$		−	3.06676e9i		−	0.323160i	0.986860	−	0.161580i	$-0.0516590\pi$
$710$		0			0
$711$	−1.09940e9			−0.114713
$712$		0			0
$713$			1.30502e9i			0.134836i
$714$		0			0
$715$	2.84090e9	+	1.76647e10i	0.290660	+	1.80732i
$716$		0			0
$717$		−	3.76893e9i		−	0.381857i
$718$		0			0
$719$	−7.17747e9			−0.720145			−0.360073	−	0.932924i	$-0.617248\pi$
$719$	−7.17747e9			−0.720145			−0.360073	+	0.932924i	$0.617248\pi$
$720$		0			0
$721$			4.35757e9i			0.432983i
$722$		0			0
$723$			1.13270e10i			1.11463i
$724$		0			0
$725$	−2.21399e10			−2.15771
$726$		0			0
$727$	1.33854e10			1.29200			0.645998	−	0.763339i	$-0.276442\pi$
$727$	1.33854e10			1.29200			0.645998	+	0.763339i	$0.276442\pi$
$728$		0			0
$729$	7.43152e9			0.710447
$730$		0			0
$731$	2.02572e9			0.191808
$732$		0			0
$733$		−	1.23188e10i		−	1.15532i	−0.816277	−	0.577661i	$-0.803965\pi$
$733$		−	1.23188e10i		−	1.15532i	0.816277	−	0.577661i	$-0.196035\pi$
$734$		0			0
$735$		−	1.11495e10i		−	1.03574i
$736$		0			0
$737$	1.46769e10			1.35051
$738$		0			0
$739$		−	3.87017e9i		−	0.352756i	−0.984323	−	0.176378i	$-0.943562\pi$
$739$		−	3.87017e9i		−	0.352756i	0.984323	−	0.176378i	$-0.0564381\pi$
$740$		0			0
$741$	−8.07873e9	+	1.29925e9i	−0.729423	+	0.117308i
$742$		0			0
$743$			1.30810e10i			1.16998i	0.811040	+	0.584990i	$0.198902\pi$
$743$			1.30810e10i			1.16998i	−0.811040	+	0.584990i	$0.801098\pi$
$744$		0			0
$745$	3.67389e9			0.325521
$746$		0			0
$747$		−	5.17729e8i		−	0.0454444i
$748$		0			0
$749$		−	1.73327e9i		−	0.150723i
$750$		0			0
$751$	−2.33046e8			−0.0200772			−0.0100386	−	0.999950i	$-0.503195\pi$
$751$	−2.33046e8			−0.0200772			−0.0100386	+	0.999950i	$0.503195\pi$
$752$		0			0
$753$	6.68632e9			0.570696
$754$		0			0
$755$	2.83203e10			2.39488
$756$		0			0
$757$	−8.89085e9			−0.744917			−0.372458	−	0.928049i	$-0.621485\pi$
$757$	−8.89085e9			−0.744917			−0.372458	+	0.928049i	$0.621485\pi$
$758$		0			0
$759$		−	1.69840e10i		−	1.40992i
$760$		0			0
$761$			1.69293e10i			1.39249i	0.717803	+	0.696247i	$0.245148\pi$
$761$			1.69293e10i			1.39249i	−0.717803	+	0.696247i	$0.754852\pi$
$762$		0			0
$763$	−1.09499e10			−0.892430
$764$		0			0
$765$			4.38883e9i			0.354433i
$766$		0			0
$767$	−1.95660e10	+	3.14667e9i	−1.56573	+	0.251807i
$768$		0			0
$769$			1.35031e10i			1.07076i	0.844611	+	0.535380i	$0.179831\pi$
$769$			1.35031e10i			1.07076i	−0.844611	+	0.535380i	$0.820169\pi$
$770$		0			0
$771$	5.72840e7			0.00450135
$772$		0			0
$773$			1.01595e10i			0.791126i	0.918439	+	0.395563i	$0.129451\pi$
$773$			1.01595e10i			0.791126i	−0.918439	+	0.395563i	$0.870549\pi$
$774$		0			0
$775$			2.93162e9i			0.226231i
$776$		0			0
$777$	−1.41417e10			−1.08150
$778$		0			0
$779$	2.04063e9			0.154662
$780$		0			0
$781$	−8.59750e9			−0.645793
$782$		0			0
$783$	−1.22698e10			−0.913425
$784$		0			0
$785$			4.05278e10i			2.99026i
$786$		0			0
$787$			2.11128e10i			1.54396i	0.635650	+	0.771978i	$0.280732\pi$
$787$			2.11128e10i			1.54396i	−0.635650	+	0.771978i	$0.719268\pi$
$788$		0			0
$789$	1.55397e10			1.12635
$790$		0			0
$791$		−	1.10758e10i		−	0.795715i
$792$		0			0
$793$	−9.96568e8	−	6.19665e9i	−0.0709661	−	0.441267i
$794$		0			0
$795$		−	5.11456e9i		−	0.361014i
$796$		0			0
$797$	1.50164e10			1.05066			0.525329	−	0.850899i	$-0.323942\pi$
$797$	1.50164e10			1.05066			0.525329	+	0.850899i	$0.323942\pi$
$798$		0			0
$799$		−	8.69614e9i		−	0.603133i
$800$		0			0
$801$			3.68149e9i			0.253111i
$802$		0			0
$803$	1.96920e10			1.34210
$804$		0			0
$805$	−2.14761e10			−1.45101
$806$		0			0
$807$	−1.25004e10			−0.837269
$808$		0			0
$809$	−1.73621e10			−1.15288			−0.576439	−	0.817140i	$-0.695558\pi$
$809$	−1.73621e10			−1.15288			−0.576439	+	0.817140i	$0.695558\pi$
$810$		0			0
$811$			2.29338e10i			1.50974i	0.655872	+	0.754872i	$0.272301\pi$
$811$			2.29338e10i			1.50974i	−0.655872	+	0.754872i	$0.727699\pi$
$812$		0			0
$813$		−	1.70552e10i		−	1.11311i
$814$		0			0
$815$	−1.64346e10			−1.06343
$816$		0			0
$817$		−	2.08182e9i		−	0.133557i
$818$		0			0
$819$	−2.21727e9	+	3.56590e8i	−0.141035	+	0.0226817i
$820$		0			0
$821$			1.35927e10i			0.857242i	0.903484	+	0.428621i	$0.141000\pi$
$821$			1.35927e10i			0.857242i	−0.903484	+	0.428621i	$0.859000\pi$
$822$		0			0
$823$	1.74580e9			0.109168			0.0545841	−	0.998509i	$-0.482617\pi$
$823$	1.74580e9			0.109168			0.0545841	+	0.998509i	$0.482617\pi$
$824$		0			0
$825$		−	3.81531e10i		−	2.36560i
$826$		0			0
$827$			1.56621e10i			0.962900i	0.876473	+	0.481450i	$0.159890\pi$
$827$			1.56621e10i			0.962900i	−0.876473	+	0.481450i	$0.840110\pi$
$828$		0			0
$829$	6.04053e9			0.368243			0.184122	−	0.982903i	$-0.441056\pi$
$829$	6.04053e9			0.368243			0.184122	+	0.982903i	$0.441056\pi$
$830$		0			0
$831$	1.50621e10			0.910503
$832$		0			0
$833$	−8.66994e9			−0.519707
$834$		0			0
$835$	−3.79098e10			−2.25345
$836$		0			0
$837$			1.62469e9i			0.0957706i
$838$		0			0
$839$		−	1.52652e10i		−	0.892348i	−0.894946	−	0.446174i	$-0.852786\pi$
$839$		−	1.52652e10i		−	0.892348i	0.894946	−	0.446174i	$-0.147214\pi$
$840$		0			0
$841$	1.82048e9			0.105536
$842$		0			0
$843$			2.47887e10i			1.42514i
$844$		0			0
$845$	9.60706e9	+	2.90958e10i	0.547762	+	1.65894i
$846$		0			0
$847$		−	1.17545e9i		−	0.0664677i
$848$		0			0
$849$	−3.19517e10			−1.79191
$850$		0			0
$851$		−	3.18328e10i		−	1.77061i
$852$		0			0
$853$			4.49630e9i			0.248047i	0.992279	+	0.124023i	$0.0395798\pi$
$853$			4.49630e9i			0.248047i	−0.992279	+	0.124023i	$0.960420\pi$
$854$		0			0
$855$	4.51038e9			0.246792
$856$		0			0
$857$	−1.87369e10			−1.01687			−0.508435	−	0.861101i	$-0.669776\pi$
$857$	−1.87369e10			−1.01687			−0.508435	+	0.861101i	$0.669776\pi$
$858$		0			0
$859$	−5.28210e9			−0.284335			−0.142168	−	0.989843i	$-0.545407\pi$
$859$	−5.28210e9			−0.284335			−0.142168	+	0.989843i	$0.545407\pi$
$860$		0			0
$861$	3.22251e9			0.172061
$862$		0			0
$863$			1.11634e10i			0.591231i	0.955307	+	0.295616i	$0.0955248\pi$
$863$			1.11634e10i			0.591231i	−0.955307	+	0.295616i	$0.904475\pi$
$864$		0			0
$865$			3.26488e8i			0.0171518i
$866$		0			0
$867$	−1.47530e9			−0.0768799
$868$		0			0
$869$		−	1.10534e10i		−	0.571382i
$870$		0			0
$871$	2.48164e10	−	3.99107e9i	1.27255	−	0.204656i
$872$		0			0
$873$			1.27427e8i			0.00648205i
$874$		0			0
$875$	−2.47350e10			−1.24820
$876$		0			0
$877$		−	1.55652e9i		−	0.0779210i	−0.999241	−	0.0389605i	$-0.987595\pi$
$877$		−	1.55652e9i		−	0.0779210i	0.999241	−	0.0389605i	$-0.0124047\pi$
$878$		0			0
$879$			1.15910e10i			0.575650i
$880$		0			0
$881$	3.77180e10			1.85837			0.929187	−	0.369611i	$-0.120509\pi$
$881$	3.77180e10			1.85837			0.929187	+	0.369611i	$0.120509\pi$
$882$		0			0
$883$	1.53832e10			0.751943			0.375972	−	0.926631i	$-0.377309\pi$
$883$	1.53832e10			0.751943			0.375972	+	0.926631i	$0.377309\pi$
$884$		0			0
$885$	6.28527e10			3.04805
$886$		0			0
$887$	1.58874e10			0.764401			0.382201	−	0.924079i	$-0.375166\pi$
$887$	1.58874e10			0.764401			0.382201	+	0.924079i	$0.375166\pi$
$888$		0			0
$889$		−	1.24083e10i		−	0.592319i
$890$		0			0
$891$		−	2.57982e10i		−	1.22185i
$892$		0			0
$893$	−8.93698e9			−0.419963
$894$		0			0
$895$		−	5.15692e10i		−	2.40442i
$896$		0			0
$897$	−4.61843e9	−	2.87174e10i	−0.213659	−	1.32853i
$898$		0			0
$899$		−	2.52517e9i		−	0.115913i
$900$		0			0
$901$	−3.97714e9			−0.181148
$902$		0			0
$903$		−	3.28754e9i		−	0.148581i
$904$		0			0
$905$			6.94896e9i			0.311638i
$906$		0			0
$907$	1.04545e10			0.465240			0.232620	−	0.972568i	$-0.425270\pi$
$907$	1.04545e10			0.465240			0.232620	+	0.972568i	$0.425270\pi$
$908$		0			0
$909$	6.10938e9			0.269789
$910$		0			0
$911$	4.40778e10			1.93155			0.965774	−	0.259386i	$-0.0835203\pi$
$911$	4.40778e10			1.93155			0.965774	+	0.259386i	$0.0835203\pi$
$912$		0			0
$913$	5.20527e9			0.226358
$914$		0			0
$915$			1.99058e10i			0.859024i
$916$		0			0
$917$			1.48757e10i			0.637067i
$918$		0			0
$919$	4.22709e10			1.79654			0.898270	−	0.439445i	$-0.144825\pi$
$919$	4.22709e10			1.79654			0.898270	+	0.439445i	$0.144825\pi$
$920$		0			0
$921$		−	1.00310e10i		−	0.423092i
$922$		0			0
$923$	−1.45371e10	+	2.33790e9i	−0.608514	+	0.0978634i
$924$		0			0
$925$		−	7.15097e10i		−	2.97077i
$926$		0			0
$927$	3.25315e9			0.134130
$928$		0			0
$929$		−	4.13059e10i		−	1.69027i	−0.534551	−	0.845136i	$-0.679519\pi$
$929$		−	4.13059e10i		−	1.69027i	0.534551	−	0.845136i	$-0.320481\pi$
$930$		0			0
$931$			8.91006e9i			0.361873i
$932$		0			0
$933$	−1.66514e10			−0.671221
$934$		0			0
$935$	−4.41255e10			−1.76542
$936$		0			0
$937$	3.77754e10			1.50010			0.750050	−	0.661382i	$-0.230030\pi$
$937$	3.77754e10			1.50010			0.750050	+	0.661382i	$0.230030\pi$
$938$		0			0
$939$	7.86879e9			0.310155
$940$		0			0
$941$			3.17351e9i			0.124159i	0.998071	+	0.0620793i	$0.0197732\pi$
$941$			3.17351e9i			0.124159i	−0.998071	+	0.0620793i	$0.980227\pi$
$942$		0			0
$943$			7.25382e9i			0.281693i
$944$		0			0
$945$	−2.67367e10			−1.03062
$946$		0			0
$947$			3.54610e10i			1.35683i	0.734677	+	0.678417i	$0.237334\pi$
$947$			3.54610e10i			1.35683i	−0.734677	+	0.678417i	$0.762666\pi$
$948$		0			0
$949$	3.32963e10	−	5.35482e9i	1.26463	−	0.203382i
$950$		0			0
$951$			5.42582e9i			0.204566i
$952$		0			0
$953$	5.38620e9			0.201585			0.100792	−	0.994907i	$-0.467862\pi$
$953$	5.38620e9			0.201585			0.100792	+	0.994907i	$0.467862\pi$
$954$		0			0
$955$		−	2.96141e10i		−	1.10024i
$956$		0			0
$957$			3.28635e10i			1.21205i
$958$		0			0
$959$	−5.15089e9			−0.188589
$960$		0			0
$961$	2.71782e10			0.987847
$962$		0			0
$963$	−1.29397e9			−0.0466910
$964$		0			0
$965$	−9.61005e10			−3.44255
$966$		0			0
$967$		−	3.69599e10i		−	1.31443i	−0.753703	−	0.657216i	$-0.771734\pi$
$967$		−	3.69599e10i		−	1.31443i	0.753703	−	0.657216i	$-0.228266\pi$
$968$		0			0
$969$		−	2.01803e10i		−	0.712515i
$970$		0			0
$971$	9.10466e9			0.319151			0.159576	−	0.987186i	$-0.448987\pi$
$971$	9.10466e9			0.319151			0.159576	+	0.987186i	$0.448987\pi$
$972$		0			0
$973$		−	2.83095e10i		−	0.985229i
$974$		0			0
$975$	−1.03749e10	−	6.45112e10i	−0.358483	−	2.22904i
$976$		0			0
$977$			1.66549e10i			0.571363i	0.958325	+	0.285681i	$0.0922199\pi$
$977$			1.66549e10i			0.571363i	−0.958325	+	0.285681i	$0.907780\pi$
$978$		0			0
$979$	−3.70139e10			−1.26074
$980$		0			0
$981$			8.17466e9i			0.276457i
$982$		0			0
$983$		−	5.63122e10i		−	1.89088i	−0.325790	−	0.945442i	$-0.605630\pi$
$983$		−	5.63122e10i		−	1.89088i	0.325790	−	0.945442i	$-0.394370\pi$
$984$		0			0
$985$	3.46002e10			1.15359
$986$		0			0
$987$	−1.41130e10			−0.467207
$988$		0			0
$989$	7.40022e9			0.243253
$990$		0			0
$991$	−3.57927e9			−0.116825			−0.0584127	−	0.998293i	$-0.518604\pi$
$991$	−3.57927e9			−0.116825			−0.0584127	+	0.998293i	$0.518604\pi$
$992$		0			0
$993$		−	3.36447e10i		−	1.09042i
$994$		0			0
$995$			6.17082e10i			1.98592i
$996$		0			0
$997$	−1.48467e10			−0.474456			−0.237228	−	0.971454i	$-0.576239\pi$
$997$	−1.48467e10			−0.474456			−0.237228	+	0.971454i	$0.576239\pi$
$998$		0			0
$999$		−	3.96304e10i		−	1.25762i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.f.a.129.5		6
4.3	odd	2		13.8.b.a.12.4	yes	6
12.11	even	2		117.8.b.b.64.3		6
13.12	even	2	inner	208.8.f.a.129.6		6
52.31	even	4		169.8.a.d.1.4		6
52.47	even	4		169.8.a.d.1.3		6
52.51	odd	2		13.8.b.a.12.3	✓	6
156.155	even	2		117.8.b.b.64.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.3	✓	6	52.51	odd	2
13.8.b.a.12.4	yes	6	4.3	odd	2
117.8.b.b.64.3		6	12.11	even	2
117.8.b.b.64.4		6	156.155	even	2
169.8.a.d.1.3		6	52.47	even	4
169.8.a.d.1.4		6	52.31	even	4
208.8.f.a.129.5		6	1.1	even	1	trivial
208.8.f.a.129.6		6	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+51.4496 q^{3} -488.312i q^{5} +616.243i q^{7} +460.056 q^{9} +4625.43i q^{11} +(1257.79 + 7820.90i) q^{13} -25123.4i q^{15} -19536.2 q^{17} +20077.3i q^{19} +31705.4i q^{21} -71368.5 q^{23} -160323. q^{25} -88850.5 q^{27} +138095. q^{29} -18285.7i q^{31} +237976. i q^{33} +300918. q^{35} +446035. i q^{37} +(64712.5 + 402382. i) q^{39} -101639. i q^{41} -103690. q^{43} -224651. i q^{45} +445129. i q^{47} +443788. q^{49} -1.00513e6 q^{51} +203578. q^{53} +2.25865e6 q^{55} +1.03297e6i q^{57} +2.50176e6i q^{59} -792320. q^{61} +283506. i q^{63} +(3.81904e6 - 614191. i) q^{65} -3.17309e6i q^{67} -3.67188e6 q^{69} +1.85875e6i q^{71} -4.25734e6i q^{73} -8.24856e6 q^{75} -2.85039e6 q^{77} -2.38970e6 q^{79} -5.57746e6 q^{81} -1.12536e6i q^{83} +9.53976e6i q^{85} +7.10495e6 q^{87} +8.00227e6i q^{89} +(-4.81957e6 + 775101. i) q^{91} -940791. i q^{93} +9.80397e6 q^{95} +276982. i q^{97} +2.12796e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 56 q^{3} - 1150 q^{9} - 5018 q^{13} + 13152 q^{17} - 27264 q^{23} - 18262 q^{25} - 194560 q^{27} + 42924 q^{29} + 546720 q^{35} - 511160 q^{39} + 1005576 q^{43} + 3246846 q^{49} - 297984 q^{51} + 1705524 q^{53}+ \cdots + 22075632 q^{95}+O(q^{100}) \) 6 * q + 56 * q^3 - 1150 * q^9 - 5018 * q^13 + 13152 * q^17 - 27264 * q^23 - 18262 * q^25 - 194560 * q^27 + 42924 * q^29 + 546720 * q^35 - 511160 * q^39 + 1005576 * q^43 + 3246846 * q^49 - 297984 * q^51 + 1705524 * q^53 + 5320576 * q^55 - 8237572 * q^61 + 7139028 * q^65 - 6017400 * q^69 - 20426072 * q^75 + 4093560 * q^77 - 15147200 * q^79 - 21318442 * q^81 + 36474240 * q^87 - 21281832 * q^91 + 22075632 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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