LMFDB - Embedded newform 208.10.f.d.129.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.129"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.8
Character	$\chi$	$=$	208.129
Dual form			208.10.f.d.129.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-161.892 q^{3} -1806.71i q^{5} +1769.55i q^{7} +6526.18 q^{9} -14165.1i q^{11} +(-52309.9 - 88702.7i) q^{13} +292493. i q^{15} +304500. q^{17} +251175. i q^{19} -286476. i q^{21} +1.51769e6 q^{23} -1.31107e6 q^{25} +2.12999e6 q^{27} -2.86696e6 q^{29} +3.38744e6i q^{31} +2.29323e6i q^{33} +3.19706e6 q^{35} +4.18575e6i q^{37} +(8.46857e6 + 1.43603e7i) q^{39} +2.26679e7i q^{41} +4.96081e6 q^{43} -1.17909e7i q^{45} +4.00883e7i q^{47} +3.72223e7 q^{49} -4.92962e7 q^{51} -6.73516e7 q^{53} -2.55923e7 q^{55} -4.06633e7i q^{57} -4.70086e7i q^{59} -4.42124e7 q^{61} +1.15484e7i q^{63} +(-1.60260e8 + 9.45087e7i) q^{65} +3.35490e7i q^{67} -2.45702e8 q^{69} +1.65059e8i q^{71} -1.28991e8i q^{73} +2.12253e8 q^{75} +2.50658e7 q^{77} +5.42215e8 q^{79} -4.73284e8 q^{81} +3.36344e8i q^{83} -5.50142e8i q^{85} +4.64139e8 q^{87} -6.49645e8i q^{89} +(1.56964e8 - 9.25647e7i) q^{91} -5.48401e8i q^{93} +4.53800e8 q^{95} -8.75479e7i q^{97} -9.24441e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) $ 32 * q + 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 - 3171556 * q^23 - 13526722 * q^25 + 3694974 * q^27 + 8833508 * q^29 + 8281126 * q^35 + 12056860 * q^39 - 89959038 * q^43 - 172344874 * q^49 - 51206562 * q^51 + 139521112 * q^53 + 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 + 604473728 * q^75 - 1666197144 * q^77 - 1165982916 * q^79 + 859636544 * q^81 + 1738691296 * q^87 + 291718146 * q^91 - 1741143356 * 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
$3$	−161.892			−1.15393			−0.576967	−	0.816767i	$-0.695764\pi$
$3$	−161.892			−1.15393			−0.576967	+	0.816767i	$0.695764\pi$
$4$		0			0
$5$		−	1806.71i		−	1.29278i	−0.763009	−	0.646388i	$-0.776279\pi$
$5$		−	1806.71i		−	1.29278i	0.763009	−	0.646388i	$-0.223721\pi$
$6$		0			0
$7$			1769.55i			0.278561i	0.990253	+	0.139281i	$0.0444790\pi$
$7$			1769.55i			0.278561i	−0.990253	+	0.139281i	$0.955521\pi$
$8$		0			0
$9$	6526.18			0.331564
$10$		0			0
$11$		−	14165.1i		−	0.291711i	−0.989306	−	0.145856i	$-0.953406\pi$
$11$		−	14165.1i		−	0.291711i	0.989306	−	0.145856i	$-0.0465935\pi$
$12$		0			0
$13$	−52309.9	−	88702.7i	−0.507970	−	0.861374i
$13$	−52309.9	−	88702.7i	−0.507970	−	0.861374i
$14$		0			0
$15$			292493.i			1.49178i
$16$		0			0
$17$	304500.			0.884233			0.442116	−	0.896958i	$-0.354228\pi$
$17$	304500.			0.884233			0.442116	+	0.896958i	$0.354228\pi$
$18$		0			0
$19$			251175.i			0.442165i	0.975255	+	0.221083i	$0.0709591\pi$
$19$			251175.i			0.442165i	−0.975255	+	0.221083i	$0.929041\pi$
$20$		0			0
$21$		−	286476.i		−	0.321441i
$22$		0			0
$23$	1.51769e6			1.13085			0.565427	−	0.824798i	$-0.308711\pi$
$23$	1.51769e6			1.13085			0.565427	+	0.824798i	$0.308711\pi$
$24$		0			0
$25$	−1.31107e6			−0.671269
$26$		0			0
$27$	2.12999e6			0.771331
$28$		0			0
$29$	−2.86696e6			−0.752715			−0.376357	−	0.926475i	$-0.622824\pi$
$29$	−2.86696e6			−0.752715			−0.376357	+	0.926475i	$0.622824\pi$
$30$		0			0
$31$			3.38744e6i			0.658785i	0.944193	+	0.329393i	$0.106844\pi$
$31$			3.38744e6i			0.658785i	−0.944193	+	0.329393i	$0.893156\pi$
$32$		0			0
$33$			2.29323e6i			0.336616i
$34$		0			0
$35$	3.19706e6			0.360117
$36$		0			0
$37$			4.18575e6i			0.367169i	0.983004	+	0.183584i	$0.0587700\pi$
$37$			4.18575e6i			0.367169i	−0.983004	+	0.183584i	$0.941230\pi$
$38$		0			0
$39$	8.46857e6	+	1.43603e7i	0.586165	+	0.993969i
$40$		0			0
$41$			2.26679e7i			1.25281i	0.779498	+	0.626404i	$0.215474\pi$
$41$			2.26679e7i			1.25281i	−0.779498	+	0.626404i	$0.784526\pi$
$42$		0			0
$43$	4.96081e6			0.221281			0.110641	−	0.993860i	$-0.464710\pi$
$43$	4.96081e6			0.221281			0.110641	+	0.993860i	$0.464710\pi$
$44$		0			0
$45$		−	1.17909e7i		−	0.428638i
$46$		0			0
$47$			4.00883e7i			1.19833i	0.800625	+	0.599165i	$0.204501\pi$
$47$			4.00883e7i			1.19833i	−0.800625	+	0.599165i	$0.795499\pi$
$48$		0			0
$49$	3.72223e7			0.922404
$50$		0			0
$51$	−4.92962e7			−1.02035
$52$		0			0
$53$	−6.73516e7			−1.17248			−0.586242	−	0.810136i	$-0.699393\pi$
$53$	−6.73516e7			−1.17248			−0.586242	+	0.810136i	$0.699393\pi$
$54$		0			0
$55$	−2.55923e7			−0.377117
$56$		0			0
$57$		−	4.06633e7i		−	0.510230i
$58$		0			0
$59$		−	4.70086e7i		−	0.505061i	−0.967589	−	0.252530i	$-0.918737\pi$
$59$		−	4.70086e7i		−	0.505061i	0.967589	−	0.252530i	$-0.0812627\pi$
$60$		0			0
$61$	−4.42124e7			−0.408846			−0.204423	−	0.978883i	$-0.565532\pi$
$61$	−4.42124e7			−0.408846			−0.204423	+	0.978883i	$0.565532\pi$
$62$		0			0
$63$			1.15484e7i			0.0923609i
$64$		0			0
$65$	−1.60260e8	+	9.45087e7i	−1.11356	+	0.656692i
$66$		0			0
$67$			3.35490e7i			0.203396i	0.994815	+	0.101698i	$0.0324276\pi$
$67$			3.35490e7i			0.203396i	−0.994815	+	0.101698i	$0.967572\pi$
$68$		0			0
$69$	−2.45702e8			−1.30493
$70$		0			0
$71$			1.65059e8i			0.770861i	0.922737	+	0.385430i	$0.125947\pi$
$71$			1.65059e8i			0.770861i	−0.922737	+	0.385430i	$0.874053\pi$
$72$		0			0
$73$		−	1.28991e8i		−	0.531627i	−0.964024	−	0.265814i	$-0.914359\pi$
$73$		−	1.28991e8i		−	0.531627i	0.964024	−	0.265814i	$-0.0856406\pi$
$74$		0			0
$75$	2.12253e8			0.774600
$76$		0			0
$77$	2.50658e7			0.0812595
$78$		0			0
$79$	5.42215e8			1.56621			0.783105	−	0.621890i	$-0.213635\pi$
$79$	5.42215e8			1.56621			0.783105	+	0.621890i	$0.213635\pi$
$80$		0			0
$81$	−4.73284e8			−1.22163
$82$		0			0
$83$			3.36344e8i			0.777914i	0.921256	+	0.388957i	$0.127165\pi$
$83$			3.36344e8i			0.777914i	−0.921256	+	0.388957i	$0.872835\pi$
$84$		0			0
$85$		−	5.50142e8i		−	1.14311i
$86$		0			0
$87$	4.64139e8			0.868584
$88$		0			0
$89$		−	6.49645e8i		−	1.09754i	−0.835973	−	0.548771i	$-0.815096\pi$
$89$		−	6.49645e8i		−	1.09754i	0.835973	−	0.548771i	$-0.184904\pi$
$90$		0			0
$91$	1.56964e8	−	9.25647e7i	0.239946	−	0.141501i
$92$		0			0
$93$		−	5.48401e8i		−	0.760195i
$94$		0			0
$95$	4.53800e8			0.571621
$96$		0			0
$97$		−	8.75479e7i		−	0.100409i	−0.998739	−	0.0502045i	$-0.984013\pi$
$97$		−	8.75479e7i		−	0.100409i	0.998739	−	0.0502045i	$-0.0159873\pi$
$98$		0			0
$99$		−	9.24441e7i		−	0.0967210i
$100$		0			0
$101$	1.36802e9			1.30812			0.654060	−	0.756443i	$-0.273065\pi$
$101$	1.36802e9			1.30812			0.654060	+	0.756443i	$0.273065\pi$
$102$		0			0
$103$	6.76715e8			0.592432			0.296216	−	0.955121i	$-0.404275\pi$
$103$	6.76715e8			0.592432			0.296216	+	0.955121i	$0.404275\pi$
$104$		0			0
$105$	−5.17579e8			−0.415552
$106$		0			0
$107$	1.28422e9			0.947136			0.473568	−	0.880757i	$-0.342966\pi$
$107$	1.28422e9			0.947136			0.473568	+	0.880757i	$0.342966\pi$
$108$		0			0
$109$		−	1.63489e8i		−	0.110935i	−0.998460	−	0.0554676i	$-0.982335\pi$
$109$		−	1.63489e8i		−	0.110935i	0.998460	−	0.0554676i	$-0.0176649\pi$
$110$		0			0
$111$		−	6.77642e8i		−	0.423688i
$112$		0			0
$113$	1.61471e9			0.931626			0.465813	−	0.884883i	$-0.345762\pi$
$113$	1.61471e9			0.931626			0.465813	+	0.884883i	$0.345762\pi$
$114$		0			0
$115$		−	2.74202e9i		−	1.46194i
$116$		0			0
$117$	−3.41383e8	−	5.78890e8i	−0.168425	−	0.285601i
$118$		0			0
$119$			5.38826e8i			0.246313i
$120$		0			0
$121$	2.15730e9			0.914905
$122$		0			0
$123$		−	3.66977e9i		−	1.44566i
$124$		0			0
$125$		−	1.16000e9i		−	0.424975i
$126$		0			0
$127$	2.84744e9			0.971267			0.485633	−	0.874163i	$-0.338589\pi$
$127$	2.84744e9			0.971267			0.485633	+	0.874163i	$0.338589\pi$
$128$		0			0
$129$	−8.03117e8			−0.255344
$130$		0			0
$131$	−4.71570e9			−1.39902			−0.699512	−	0.714621i	$-0.746599\pi$
$131$	−4.71570e9			−1.39902			−0.699512	+	0.714621i	$0.746599\pi$
$132$		0			0
$133$	−4.44465e8			−0.123170
$134$		0			0
$135$		−	3.84827e9i		−	0.997158i
$136$		0			0
$137$		−	4.29974e9i		−	1.04280i	−0.853313	−	0.521399i	$-0.825411\pi$
$137$		−	4.29974e9i		−	1.04280i	0.853313	−	0.521399i	$-0.174589\pi$
$138$		0			0
$139$	−3.26473e9			−0.741790			−0.370895	−	0.928675i	$-0.620949\pi$
$139$	−3.26473e9			−0.741790			−0.370895	+	0.928675i	$0.620949\pi$
$140$		0			0
$141$		−	6.48999e9i		−	1.38280i
$142$		0			0
$143$	−1.25649e9	+	7.40976e8i	−0.251273	+	0.148181i
$144$		0			0
$145$			5.17976e9i			0.973092i
$146$		0			0
$147$	−6.02601e9			−1.06439
$148$		0			0
$149$		−	3.06111e9i		−	0.508793i	−0.967100	−	0.254397i	$-0.918123\pi$
$149$		−	3.06111e9i		−	0.508793i	0.967100	−	0.254397i	$-0.0818769\pi$
$150$		0			0
$151$			4.22959e9i			0.662067i	0.943619	+	0.331033i	$0.107397\pi$
$151$			4.22959e9i			0.662067i	−0.943619	+	0.331033i	$0.892603\pi$
$152$		0			0
$153$	1.98722e9			0.293180
$154$		0			0
$155$	6.12012e9			0.851662
$156$		0			0
$157$	6.94901e9			0.912798			0.456399	−	0.889775i	$-0.349139\pi$
$157$	6.94901e9			0.912798			0.456399	+	0.889775i	$0.349139\pi$
$158$		0			0
$159$	1.09037e10			1.35297
$160$		0			0
$161$			2.68562e9i			0.315012i
$162$		0			0
$163$		−	1.17678e10i		−	1.30572i	−0.757477	−	0.652862i	$-0.773568\pi$
$163$		−	1.17678e10i		−	1.30572i	0.757477	−	0.652862i	$-0.226432\pi$
$164$		0			0
$165$	4.14319e9			0.435169
$166$		0			0
$167$		−	9.00553e9i		−	0.895953i	−0.894045	−	0.447976i	$-0.852145\pi$
$167$		−	9.00553e9i		−	0.895953i	0.894045	−	0.447976i	$-0.147855\pi$
$168$		0			0
$169$	−5.13186e9	+	9.28006e9i	−0.483932	+	0.875106i
$170$		0			0
$171$			1.63921e9i			0.146606i
$172$		0			0
$173$	1.58019e10			1.34122			0.670612	−	0.741808i	$-0.266031\pi$
$173$	1.58019e10			1.34122			0.670612	+	0.741808i	$0.266031\pi$
$174$		0			0
$175$		−	2.32000e9i		−	0.186990i
$176$		0			0
$177$			7.61034e9i			0.582807i
$178$		0			0
$179$	−2.31701e10			−1.68690			−0.843449	−	0.537209i	$-0.819479\pi$
$179$	−2.31701e10			−1.68690			−0.843449	+	0.537209i	$0.819479\pi$
$180$		0			0
$181$	8.68481e9			0.601460			0.300730	−	0.953709i	$-0.402770\pi$
$181$	8.68481e9			0.601460			0.300730	+	0.953709i	$0.402770\pi$
$182$		0			0
$183$	7.15766e9			0.471782
$184$		0			0
$185$	7.56243e9			0.474667
$186$		0			0
$187$		−	4.31327e9i		−	0.257941i
$188$		0			0
$189$			3.76912e9i			0.214863i
$190$		0			0
$191$	−1.49221e10			−0.811299			−0.405649	−	0.914029i	$-0.632955\pi$
$191$	−1.49221e10			−0.811299			−0.405649	+	0.914029i	$0.632955\pi$
$192$		0			0
$193$		−	1.29190e10i		−	0.670227i	−0.942178	−	0.335114i	$-0.891225\pi$
$193$		−	1.29190e10i		−	0.670227i	0.942178	−	0.335114i	$-0.108775\pi$
$194$		0			0
$195$	2.59449e10	−	1.53002e10i	1.28498	−	0.757779i
$196$		0			0
$197$		−	1.44295e10i		−	0.682580i	−0.939958	−	0.341290i	$-0.889136\pi$
$197$		−	1.44295e10i		−	0.682580i	0.939958	−	0.341290i	$-0.110864\pi$
$198$		0			0
$199$	−1.82419e10			−0.824576			−0.412288	−	0.911054i	$-0.635270\pi$
$199$	−1.82419e10			−0.824576			−0.412288	+	0.911054i	$0.635270\pi$
$200$		0			0
$201$		−	5.43133e9i		−	0.234706i
$202$		0			0
$203$		−	5.07322e9i		−	0.209677i
$204$		0			0
$205$	4.09544e10			1.61960
$206$		0			0
$207$	9.90469e9			0.374951
$208$		0			0
$209$	3.55792e9			0.128985
$210$		0			0
$211$	2.99022e10			1.03856			0.519281	−	0.854603i	$-0.326200\pi$
$211$	2.99022e10			1.03856			0.519281	+	0.854603i	$0.326200\pi$
$212$		0			0
$213$		−	2.67218e10i		−	0.889523i
$214$		0			0
$215$		−	8.96273e9i		−	0.286067i
$216$		0			0
$217$	−5.99423e9			−0.183512
$218$		0			0
$219$			2.08827e10i			0.613463i
$220$		0			0
$221$	−1.59283e10	−	2.70100e10i	−0.449164	−	0.761655i
$222$		0			0
$223$		−	1.11106e10i		−	0.300860i	−0.988621	−	0.150430i	$-0.951934\pi$
$223$		−	1.11106e10i		−	0.300860i	0.988621	−	0.150430i	$-0.0480658\pi$
$224$		0			0
$225$	−8.55629e9			−0.222569
$226$		0			0
$227$		−	5.88854e9i		−	0.147195i	−0.997288	−	0.0735973i	$-0.976552\pi$
$227$		−	5.88854e9i		−	0.147195i	0.997288	−	0.0735973i	$-0.0234479\pi$
$228$		0			0
$229$		−	4.86793e10i		−	1.16973i	−0.811132	−	0.584863i	$-0.801148\pi$
$229$		−	4.86793e10i		−	1.16973i	0.811132	−	0.584863i	$-0.198852\pi$
$230$		0			0
$231$	−4.05797e9			−0.0937681
$232$		0			0
$233$	−7.35622e10			−1.63513			−0.817566	−	0.575834i	$-0.804677\pi$
$233$	−7.35622e10			−1.63513			−0.817566	+	0.575834i	$0.804677\pi$
$234$		0			0
$235$	7.24278e10			1.54917
$236$		0			0
$237$	−8.77806e10			−1.80730
$238$		0			0
$239$		−	7.73385e10i		−	1.53322i	−0.642111	−	0.766611i	$-0.721941\pi$
$239$		−	7.73385e10i		−	1.53322i	0.642111	−	0.766611i	$-0.278059\pi$
$240$		0			0
$241$			4.64659e10i			0.887274i	0.896207	+	0.443637i	$0.146312\pi$
$241$			4.64659e10i			0.887274i	−0.896207	+	0.443637i	$0.853688\pi$
$242$		0			0
$243$	3.46966e10			0.638349
$244$		0			0
$245$		−	6.72499e10i		−	1.19246i
$246$		0			0
$247$	2.22799e10	−	1.31389e10i	0.380870	−	0.224607i
$248$		0			0
$249$		−	5.44515e10i		−	0.897662i
$250$		0			0
$251$	1.15393e10			0.183506			0.0917528	−	0.995782i	$-0.470753\pi$
$251$	1.15393e10			0.183506			0.0917528	+	0.995782i	$0.470753\pi$
$252$		0			0
$253$		−	2.14982e10i		−	0.329883i
$254$		0			0
$255$			8.90639e10i			1.31908i
$256$		0			0
$257$	1.17405e9			0.0167875			0.00839375	−	0.999965i	$-0.497328\pi$
$257$	1.17405e9			0.0167875			0.00839375	+	0.999965i	$0.497328\pi$
$258$		0			0
$259$	−7.40688e9			−0.102279
$260$		0			0
$261$	−1.87103e10			−0.249573
$262$		0			0
$263$	4.19093e9			0.0540143			0.0270072	−	0.999635i	$-0.491402\pi$
$263$	4.19093e9			0.0540143			0.0270072	+	0.999635i	$0.491402\pi$
$264$		0			0
$265$			1.21685e11i			1.51576i
$266$		0			0
$267$			1.05173e11i			1.26649i
$268$		0			0
$269$	5.17080e10			0.602106			0.301053	−	0.953607i	$-0.402662\pi$
$269$	5.17080e10			0.602106			0.301053	+	0.953607i	$0.402662\pi$
$270$		0			0
$271$			1.23417e11i			1.39000i	0.719010	+	0.695000i	$0.244596\pi$
$271$			1.23417e11i			1.39000i	−0.719010	+	0.695000i	$0.755404\pi$
$272$		0			0
$273$	−2.54112e10	+	1.49855e10i	−0.276881	+	0.163283i
$274$		0			0
$275$			1.85715e10i			0.195817i
$276$		0			0
$277$	1.31860e11			1.34572			0.672860	−	0.739770i	$-0.265066\pi$
$277$	1.31860e11			1.34572			0.672860	+	0.739770i	$0.265066\pi$
$278$		0			0
$279$			2.21070e10i			0.218430i
$280$		0			0
$281$			9.33673e10i			0.893339i	0.894699	+	0.446669i	$0.147390\pi$
$281$			9.33673e10i			0.893339i	−0.894699	+	0.446669i	$0.852610\pi$
$282$		0			0
$283$	−1.23637e9			−0.0114580			−0.00572900	−	0.999984i	$-0.501824\pi$
$283$	−1.23637e9			−0.0114580			−0.00572900	+	0.999984i	$0.501824\pi$
$284$		0			0
$285$	−7.34668e10			−0.659613
$286$		0			0
$287$	−4.01120e10			−0.348984
$288$		0			0
$289$	−2.58679e10			−0.218133
$290$		0			0
$291$			1.41733e10i			0.115865i
$292$		0			0
$293$			1.78680e11i			1.41636i	0.706034	+	0.708178i	$0.250483\pi$
$293$			1.78680e11i			1.41636i	−0.706034	+	0.708178i	$0.749517\pi$
$294$		0			0
$295$	−8.49309e10			−0.652930
$296$		0			0
$297$		−	3.01716e10i		−	0.225006i
$298$		0			0
$299$	−7.93900e10	−	1.34623e11i	−0.574441	−	0.974089i
$300$		0			0
$301$			8.77838e9i			0.0616403i
$302$		0			0
$303$	−2.21473e11			−1.50948
$304$		0			0
$305$			7.98790e10i			0.528547i
$306$		0			0
$307$			2.21356e11i			1.42223i	0.703077	+	0.711114i	$0.251809\pi$
$307$			2.21356e11i			1.42223i	−0.703077	+	0.711114i	$0.748191\pi$
$308$		0			0
$309$	−1.09555e11			−0.683628
$310$		0			0
$311$	7.46241e9			0.0452332			0.0226166	−	0.999744i	$-0.492800\pi$
$311$	7.46241e9			0.0452332			0.0226166	+	0.999744i	$0.492800\pi$
$312$		0			0
$313$	6.44761e10			0.379708			0.189854	−	0.981812i	$-0.439199\pi$
$313$	6.44761e10			0.379708			0.189854	+	0.981812i	$0.439199\pi$
$314$		0			0
$315$	2.08646e10			0.119402
$316$		0			0
$317$			2.84348e11i			1.58155i	0.612104	+	0.790777i	$0.290323\pi$
$317$			2.84348e11i			1.58155i	−0.612104	+	0.790777i	$0.709677\pi$
$318$		0			0
$319$			4.06108e10i			0.219575i
$320$		0			0
$321$	−2.07906e11			−1.09293
$322$		0			0
$323$			7.64826e10i			0.390977i
$324$		0			0
$325$	6.85820e10	+	1.16296e11i	0.340985	+	0.578214i
$326$		0			0
$327$			2.64676e10i			0.128012i
$328$		0			0
$329$	−7.09380e10			−0.333809
$330$		0			0
$331$		−	1.27498e11i		−	0.583816i	−0.956446	−	0.291908i	$-0.905710\pi$
$331$		−	1.27498e11i		−	0.583816i	0.956446	−	0.291908i	$-0.0942901\pi$
$332$		0			0
$333$			2.73170e10i			0.121740i
$334$		0			0
$335$	6.06133e10			0.262946
$336$		0			0
$337$	6.95680e10			0.293816			0.146908	−	0.989150i	$-0.453068\pi$
$337$	6.95680e10			0.293816			0.146908	+	0.989150i	$0.453068\pi$
$338$		0			0
$339$	−2.61409e11			−1.07503
$340$		0			0
$341$	4.79835e10			0.192175
$342$		0			0
$343$			1.37274e11i			0.535507i
$344$		0			0
$345$			4.43912e11i			1.68698i
$346$		0			0
$347$	−1.97799e11			−0.732388			−0.366194	−	0.930538i	$-0.619339\pi$
$347$	−1.97799e11			−0.732388			−0.366194	+	0.930538i	$0.619339\pi$
$348$		0			0
$349$		−	2.49812e11i		−	0.901361i	−0.892685	−	0.450681i	$-0.851181\pi$
$349$		−	2.49812e11i		−	0.901361i	0.892685	−	0.450681i	$-0.148819\pi$
$350$		0			0
$351$	−1.11420e11	−	1.88936e11i	−0.391813	−	0.664405i
$352$		0			0
$353$		−	3.74342e11i		−	1.28316i	−0.767055	−	0.641582i	$-0.778278\pi$
$353$		−	3.74342e11i		−	1.28316i	0.767055	−	0.641582i	$-0.221722\pi$
$354$		0			0
$355$	2.98213e11			0.996550
$356$		0			0
$357$		−	8.72319e10i		−	0.284229i
$358$		0			0
$359$			4.56957e11i			1.45195i	0.687723	+	0.725973i	$0.258610\pi$
$359$			4.56957e11i			1.45195i	−0.687723	+	0.725973i	$0.741390\pi$
$360$		0			0
$361$	2.59599e11			0.804490
$362$		0			0
$363$	−3.49250e11			−1.05574
$364$		0			0
$365$	−2.33050e11			−0.687275
$366$		0			0
$367$	−5.03992e11			−1.45019			−0.725097	−	0.688647i	$-0.758205\pi$
$367$	−5.03992e11			−1.45019			−0.725097	+	0.688647i	$0.758205\pi$
$368$		0			0
$369$			1.47935e11i			0.415387i
$370$		0			0
$371$		−	1.19182e11i		−	0.326608i
$372$		0			0
$373$	−9.89063e10			−0.264566			−0.132283	−	0.991212i	$-0.542231\pi$
$373$	−9.89063e10			−0.264566			−0.132283	+	0.991212i	$0.542231\pi$
$374$		0			0
$375$			1.87796e11i			0.490394i
$376$		0			0
$377$	1.49970e11	+	2.54307e11i	0.382357	+	0.648369i
$378$		0			0
$379$		−	3.53892e11i		−	0.881037i	−0.897744	−	0.440519i	$-0.854795\pi$
$379$		−	3.53892e11i		−	0.881037i	0.897744	−	0.440519i	$-0.145205\pi$
$380$		0			0
$381$	−4.60980e11			−1.12078
$382$		0			0
$383$		−	3.25704e11i		−	0.773443i	−0.922197	−	0.386722i	$-0.873607\pi$
$383$		−	3.25704e11i		−	0.773443i	0.922197	−	0.386722i	$-0.126393\pi$
$384$		0			0
$385$		−	4.52867e10i		−	0.105050i
$386$		0			0
$387$	3.23751e10			0.0733689
$388$		0			0
$389$	6.84019e11			1.51459			0.757295	−	0.653073i	$-0.226521\pi$
$389$	6.84019e11			1.51459			0.757295	+	0.653073i	$0.226521\pi$
$390$		0			0
$391$	4.62135e11			0.999939
$392$		0			0
$393$	7.63436e11			1.61438
$394$		0			0
$395$		−	9.79625e11i		−	2.02476i
$396$		0			0
$397$			1.23574e11i			0.249672i	0.992177	+	0.124836i	$0.0398405\pi$
$397$			1.23574e11i			0.249672i	−0.992177	+	0.124836i	$0.960160\pi$
$398$		0			0
$399$	7.19556e10			0.142130
$400$		0			0
$401$		−	1.00477e12i		−	1.94052i	−0.242068	−	0.970259i	$-0.577826\pi$
$401$		−	1.00477e12i		−	1.94052i	0.242068	−	0.970259i	$-0.422174\pi$
$402$		0			0
$403$	3.00475e11	−	1.77197e11i	0.567461	−	0.334644i
$404$		0			0
$405$			8.55087e11i			1.57929i
$406$		0			0
$407$	5.92917e10			0.107107
$408$		0			0
$409$		−	4.94666e11i		−	0.874093i	−0.899439	−	0.437046i	$-0.856025\pi$
$409$		−	4.94666e11i		−	0.874093i	0.899439	−	0.437046i	$-0.143975\pi$
$410$		0			0
$411$			6.96096e11i			1.20332i
$412$		0			0
$413$	8.31839e10			0.140690
$414$		0			0
$415$	6.07675e11			1.00567
$416$		0			0
$417$	5.28536e11			0.855977
$418$		0			0
$419$	−1.92154e11			−0.304570			−0.152285	−	0.988337i	$-0.548663\pi$
$419$	−1.92154e11			−0.304570			−0.152285	+	0.988337i	$0.548663\pi$
$420$		0			0
$421$			7.60538e11i			1.17992i	0.807433	+	0.589959i	$0.200856\pi$
$421$			7.60538e11i			1.17992i	−0.807433	+	0.589959i	$0.799144\pi$
$422$		0			0
$423$			2.61623e11i			0.397324i
$424$		0			0
$425$	−3.99221e11			−0.593558
$426$		0			0
$427$		−	7.82360e10i		−	0.113889i
$428$		0			0
$429$	2.03416e11	−	1.19958e11i	0.289952	−	0.170991i
$430$		0			0
$431$			1.21274e12i			1.69286i	0.532499	+	0.846431i	$0.321253\pi$
$431$			1.21274e12i			1.69286i	−0.532499	+	0.846431i	$0.678747\pi$
$432$		0			0
$433$	−1.23847e12			−1.69313			−0.846567	−	0.532282i	$-0.821335\pi$
$433$	−1.23847e12			−1.69313			−0.846567	+	0.532282i	$0.821335\pi$
$434$		0			0
$435$		−	8.38564e11i		−	1.12288i
$436$		0			0
$437$			3.81205e11i			0.500025i
$438$		0			0
$439$	1.06357e12			1.36671			0.683355	−	0.730086i	$-0.260520\pi$
$439$	1.06357e12			1.36671			0.683355	+	0.730086i	$0.260520\pi$
$440$		0			0
$441$	2.42919e11			0.305836
$442$		0			0
$443$	1.01457e11			0.125160			0.0625800	−	0.998040i	$-0.480067\pi$
$443$	1.01457e11			0.125160			0.0625800	+	0.998040i	$0.480067\pi$
$444$		0			0
$445$	−1.17372e12			−1.41888
$446$		0			0
$447$			4.95571e11i			0.587114i
$448$		0			0
$449$		−	1.00840e12i		−	1.17091i	−0.810704	−	0.585456i	$-0.800915\pi$
$449$		−	1.00840e12i		−	1.17091i	0.810704	−	0.585456i	$-0.199085\pi$
$450$		0			0
$451$	3.21094e11			0.365458
$452$		0			0
$453$		−	6.84738e11i		−	0.763981i
$454$		0			0
$455$	−1.67238e11	−	2.83588e11i	−0.182929	−	0.310196i
$456$		0			0
$457$		−	1.19455e12i		−	1.28109i	−0.767919	−	0.640547i	$-0.778708\pi$
$457$		−	1.19455e12i		−	1.28109i	0.767919	−	0.640547i	$-0.221292\pi$
$458$		0			0
$459$	6.48581e11			0.682036
$460$		0			0
$461$		−	1.75517e12i		−	1.80995i	−0.425468	−	0.904974i	$-0.639890\pi$
$461$		−	1.75517e12i		−	1.80995i	0.425468	−	0.904974i	$-0.360110\pi$
$462$		0			0
$463$			1.22344e12i			1.23728i	0.785675	+	0.618639i	$0.212316\pi$
$463$			1.22344e12i			1.23728i	−0.785675	+	0.618639i	$0.787684\pi$
$464$		0			0
$465$	−9.90801e11			−0.982762
$466$		0			0
$467$	2.48116e11			0.241396			0.120698	−	0.992689i	$-0.461487\pi$
$467$	2.48116e11			0.241396			0.120698	+	0.992689i	$0.461487\pi$
$468$		0			0
$469$	−5.93665e10			−0.0566583
$470$		0			0
$471$	−1.12499e12			−1.05331
$472$		0			0
$473$		−	7.02704e10i		−	0.0645502i
$474$		0			0
$475$		−	3.29308e11i		−	0.296812i
$476$		0			0
$477$	−4.39549e11			−0.388754
$478$		0			0
$479$		−	1.07902e11i		−	0.0936524i	−0.998903	−	0.0468262i	$-0.985089\pi$
$479$		−	1.07902e11i		−	0.0936524i	0.998903	−	0.0468262i	$-0.0149107\pi$
$480$		0			0
$481$	3.71288e11	−	2.18956e11i	0.316270	−	0.186511i
$482$		0			0
$483$		−	4.34781e11i		−	0.363503i
$484$		0			0
$485$	−1.58174e11			−0.129806
$486$		0			0
$487$		−	6.30039e11i		−	0.507560i	−0.967262	−	0.253780i	$-0.918326\pi$
$487$		−	6.30039e11i		−	0.507560i	0.967262	−	0.253780i	$-0.0816739\pi$
$488$		0			0
$489$			1.90512e12i			1.50672i
$490$		0			0
$491$	1.94699e12			1.51181			0.755906	−	0.654680i	$-0.227197\pi$
$491$	1.94699e12			1.51181			0.755906	+	0.654680i	$0.227197\pi$
$492$		0			0
$493$	−8.72988e11			−0.665575
$494$		0			0
$495$	−1.67020e11			−0.125039
$496$		0			0
$497$	−2.92079e11			−0.214732
$498$		0			0
$499$			1.56005e11i			0.112638i	0.998413	+	0.0563191i	$0.0179364\pi$
$499$			1.56005e11i			0.112638i	−0.998413	+	0.0563191i	$0.982064\pi$
$500$		0			0
$501$			1.45793e12i			1.03387i
$502$		0			0
$503$	−3.45707e11			−0.240797			−0.120399	−	0.992726i	$-0.538417\pi$
$503$	−3.45707e11			−0.240797			−0.120399	+	0.992726i	$0.538417\pi$
$504$		0			0
$505$		−	2.47162e12i		−	1.69110i
$506$		0			0
$507$	8.30809e11	−	1.50237e12i	0.558426	−	1.00981i
$508$		0			0
$509$			4.73483e11i			0.312662i	0.987705	+	0.156331i	$0.0499666\pi$
$509$			4.73483e11i			0.312662i	−0.987705	+	0.156331i	$0.950033\pi$
$510$		0			0
$511$	2.28256e11			0.148091
$512$		0			0
$513$			5.35000e11i			0.341056i
$514$		0			0
$515$		−	1.22263e12i		−	0.765882i
$516$		0			0
$517$	5.67855e11			0.349567
$518$		0			0
$519$	−2.55821e12			−1.54768
$520$		0			0
$521$	1.57930e12			0.939063			0.469532	−	0.882916i	$-0.344423\pi$
$521$	1.57930e12			0.939063			0.469532	+	0.882916i	$0.344423\pi$
$522$		0			0
$523$	7.93825e11			0.463946			0.231973	−	0.972722i	$-0.425482\pi$
$523$	7.93825e11			0.463946			0.231973	+	0.972722i	$0.425482\pi$
$524$		0			0
$525$			3.75591e11i			0.215774i
$526$		0			0
$527$			1.03147e12i			0.582520i
$528$		0			0
$529$	5.02220e11			0.278832
$530$		0			0
$531$		−	3.06787e11i		−	0.167460i
$532$		0			0
$533$	2.01071e12	−	1.18576e12i	1.07914	−	0.636390i
$534$		0			0
$535$		−	2.32021e12i		−	1.22443i
$536$		0			0
$537$	3.75106e12			1.94657
$538$		0			0
$539$		−	5.27259e11i		−	0.269076i
$540$		0			0
$541$		−	2.25361e12i		−	1.13107i	−0.824723	−	0.565537i	$-0.808669\pi$
$541$		−	2.25361e12i		−	1.13107i	0.824723	−	0.565537i	$-0.191331\pi$
$542$		0			0
$543$	−1.40601e12			−0.694045
$544$		0			0
$545$	−2.95377e11			−0.143414
$546$		0			0
$547$	3.39066e11			0.161935			0.0809676	−	0.996717i	$-0.474199\pi$
$547$	3.39066e11			0.161935			0.0809676	+	0.996717i	$0.474199\pi$
$548$		0			0
$549$	−2.88538e11			−0.135559
$550$		0			0
$551$		−	7.20108e11i		−	0.332825i
$552$		0			0
$553$			9.59475e11i			0.436285i
$554$		0			0
$555$	−1.22430e12			−0.547734
$556$		0			0
$557$			2.61638e12i			1.15173i	0.817543	+	0.575867i	$0.195336\pi$
$557$			2.61638e12i			1.15173i	−0.817543	+	0.575867i	$0.804664\pi$
$558$		0			0
$559$	−2.59499e11	−	4.40037e11i	−0.112404	−	0.190606i
$560$		0			0
$561$			6.98287e11i			0.297647i
$562$		0			0
$563$	−8.01952e9			−0.00336404			−0.00168202	−	0.999999i	$-0.500535\pi$
$563$	−8.01952e9			−0.00336404			−0.00168202	+	0.999999i	$0.500535\pi$
$564$		0			0
$565$		−	2.91731e12i		−	1.20438i
$566$		0			0
$567$		−	8.37498e11i		−	0.340299i
$568$		0			0
$569$	4.68631e12			1.87424			0.937122	−	0.349002i	$-0.113479\pi$
$569$	4.68631e12			1.87424			0.937122	+	0.349002i	$0.113479\pi$
$570$		0			0
$571$	3.72776e12			1.46752			0.733762	−	0.679407i	$-0.237763\pi$
$571$	3.72776e12			1.46752			0.733762	+	0.679407i	$0.237763\pi$
$572$		0			0
$573$	2.41578e12			0.936185
$574$		0			0
$575$	−1.98980e12			−0.759108
$576$		0			0
$577$		−	2.38246e12i		−	0.894818i	−0.894329	−	0.447409i	$-0.852347\pi$
$577$		−	2.38246e12i		−	0.894818i	0.894329	−	0.447409i	$-0.147653\pi$
$578$		0			0
$579$			2.09149e12i			0.773398i
$580$		0			0
$581$	−5.95176e11			−0.216697
$582$		0			0
$583$			9.54044e11i			0.342027i
$584$		0			0
$585$	−1.04589e12	+	6.16781e11i	−0.369218	+	0.217736i
$586$		0			0
$587$		−	3.77574e12i		−	1.31259i	−0.754503	−	0.656297i	$-0.772122\pi$
$587$		−	3.77574e12i		−	1.31259i	0.754503	−	0.656297i	$-0.227878\pi$
$588$		0			0
$589$	−8.50840e11			−0.291292
$590$		0			0
$591$			2.33603e12i			0.787652i
$592$		0			0
$593$			2.93920e12i			0.976073i	0.872823	+	0.488037i	$0.162287\pi$
$593$			2.93920e12i			0.976073i	−0.872823	+	0.488037i	$0.837713\pi$
$594$		0			0
$595$	9.73502e11			0.318427
$596$		0			0
$597$	2.95322e12			0.951507
$598$		0			0
$599$	4.70444e12			1.49309			0.746547	−	0.665333i	$-0.231711\pi$
$599$	4.70444e12			1.49309			0.746547	+	0.665333i	$0.231711\pi$
$600$		0			0
$601$	−5.14763e12			−1.60943			−0.804716	−	0.593660i	$-0.797682\pi$
$601$	−5.14763e12			−1.60943			−0.804716	+	0.593660i	$0.797682\pi$
$602$		0			0
$603$			2.18947e11i			0.0674390i
$604$		0			0
$605$		−	3.89761e12i		−	1.18277i
$606$		0			0
$607$	−5.14938e12			−1.53959			−0.769797	−	0.638289i	$-0.779642\pi$
$607$	−5.14938e12			−1.53959			−0.769797	+	0.638289i	$0.779642\pi$
$608$		0			0
$609$			8.21316e11i			0.241954i
$610$		0			0
$611$	3.55594e12	−	2.09701e12i	1.03221	−	0.608717i
$612$		0			0
$613$			2.93793e12i			0.840366i	0.907439	+	0.420183i	$0.138034\pi$
$613$			2.93793e12i			0.840366i	−0.907439	+	0.420183i	$0.861966\pi$
$614$		0			0
$615$	−6.63021e12			−1.86891
$616$		0			0
$617$			3.45557e12i			0.959923i	0.877290	+	0.479962i	$0.159349\pi$
$617$			3.45557e12i			0.959923i	−0.877290	+	0.479962i	$0.840651\pi$
$618$		0			0
$619$		−	5.81299e12i		−	1.59145i	−0.605661	−	0.795723i	$-0.707091\pi$
$619$		−	5.81299e12i		−	1.59145i	0.605661	−	0.795723i	$-0.292909\pi$
$620$		0			0
$621$	3.23266e12			0.872263
$622$		0			0
$623$	1.14958e12			0.305733
$624$		0			0
$625$	−4.65647e12			−1.22067
$626$		0			0
$627$	−5.76001e11			−0.148840
$628$		0			0
$629$			1.27456e12i			0.324663i
$630$		0			0
$631$			7.62901e12i			1.91574i	0.287205	+	0.957869i	$0.407274\pi$
$631$			7.62901e12i			1.91574i	−0.287205	+	0.957869i	$0.592726\pi$
$632$		0			0
$633$	−4.84095e12			−1.19843
$634$		0			0
$635$		−	5.14450e12i		−	1.25563i
$636$		0			0
$637$	−1.94709e12	−	3.30172e12i	−0.468554	−	0.794535i
$638$		0			0
$639$			1.07720e12i			0.255590i
$640$		0			0
$641$	−3.10612e12			−0.726704			−0.363352	−	0.931652i	$-0.618368\pi$
$641$	−3.10612e12			−0.726704			−0.363352	+	0.931652i	$0.618368\pi$
$642$		0			0
$643$			5.55594e10i			0.0128176i	0.999979	+	0.00640882i	$0.00204000\pi$
$643$			5.55594e10i			0.0128176i	−0.999979	+	0.00640882i	$0.997960\pi$
$644$		0			0
$645$			1.45100e12i			0.330102i
$646$		0			0
$647$	−6.35892e12			−1.42664			−0.713319	−	0.700839i	$-0.752809\pi$
$647$	−6.35892e12			−1.42664			−0.713319	+	0.700839i	$0.752809\pi$
$648$		0			0
$649$	−6.65883e11			−0.147332
$650$		0			0
$651$	9.70421e11			0.211761
$652$		0			0
$653$	2.16803e12			0.466611			0.233306	−	0.972403i	$-0.425046\pi$
$653$	2.16803e12			0.466611			0.233306	+	0.972403i	$0.425046\pi$
$654$		0			0
$655$			8.51989e12i			1.80862i
$656$		0			0
$657$		−	8.41820e11i		−	0.176269i
$658$		0			0
$659$	−6.43123e12			−1.32834			−0.664171	−	0.747581i	$-0.731215\pi$
$659$	−6.43123e12			−1.32834			−0.664171	+	0.747581i	$0.731215\pi$
$660$		0			0
$661$			2.23637e12i			0.455656i	0.973701	+	0.227828i	$0.0731624\pi$
$661$			2.23637e12i			0.455656i	−0.973701	+	0.227828i	$0.926838\pi$
$662$		0			0
$663$	2.57868e12	+	4.37271e12i	0.518306	+	0.878900i
$664$		0			0
$665$			8.03020e11i			0.159231i
$666$		0			0
$667$	−4.35115e12			−0.851211
$668$		0			0
$669$			1.79872e12i			0.347173i
$670$		0			0
$671$			6.26275e11i			0.119265i
$672$		0			0
$673$	2.78081e12			0.522521			0.261261	−	0.965268i	$-0.415862\pi$
$673$	2.78081e12			0.522521			0.261261	+	0.965268i	$0.415862\pi$
$674$		0			0
$675$	−2.79257e12			−0.517771
$676$		0			0
$677$	4.06401e12			0.743543			0.371771	−	0.928324i	$-0.378751\pi$
$677$	4.06401e12			0.743543			0.371771	+	0.928324i	$0.378751\pi$
$678$		0			0
$679$	1.54920e11			0.0279701
$680$		0			0
$681$			9.53311e11i			0.169853i
$682$		0			0
$683$			1.46452e12i			0.257515i	0.991676	+	0.128758i	$0.0410989\pi$
$683$			1.46452e12i			0.257515i	−0.991676	+	0.128758i	$0.958901\pi$
$684$		0			0
$685$	−7.76838e12			−1.34810
$686$		0			0
$687$			7.88081e12i			1.34979i
$688$		0			0
$689$	3.52315e12	+	5.97427e12i	0.595587	+	1.00995i
$690$		0			0
$691$			3.12626e12i			0.521643i	0.965387	+	0.260822i	$0.0839935\pi$
$691$			3.12626e12i			0.521643i	−0.965387	+	0.260822i	$0.916007\pi$
$692$		0			0
$693$	1.63584e11			0.0269427
$694$		0			0
$695$			5.89842e12i			0.958968i
$696$		0			0
$697$			6.90238e12i			1.10777i
$698$		0			0
$699$	1.19092e13			1.88684
$700$		0			0
$701$	−8.95207e12			−1.40021			−0.700104	−	0.714041i	$-0.746863\pi$
$701$	−8.95207e12			−1.40021			−0.700104	+	0.714041i	$0.746863\pi$
$702$		0			0
$703$	−1.05135e12			−0.162349
$704$		0			0
$705$	−1.17255e13			−1.78764
$706$		0			0
$707$			2.42078e12i			0.364391i
$708$		0			0
$709$		−	9.58929e12i		−	1.42521i	−0.701566	−	0.712604i	$-0.747516\pi$
$709$		−	9.58929e12i		−	1.42521i	0.701566	−	0.712604i	$-0.252484\pi$
$710$		0			0
$711$	3.53859e12			0.519299
$712$		0			0
$713$			5.14107e12i			0.744991i
$714$		0			0
$715$	1.33873e12	+	2.27010e12i	0.191564	+	0.324839i
$716$		0			0
$717$			1.25205e13i			1.76924i
$718$		0			0
$719$	−7.52612e12			−1.05025			−0.525123	−	0.851026i	$-0.675981\pi$
$719$	−7.52612e12			−1.05025			−0.525123	+	0.851026i	$0.675981\pi$
$720$		0			0
$721$			1.19748e12i			0.165029i
$722$		0			0
$723$		−	7.52248e12i		−	1.02386i
$724$		0			0
$725$	3.75879e12			0.505274
$726$		0			0
$727$	1.26694e13			1.68210			0.841052	−	0.540955i	$-0.181937\pi$
$727$	1.26694e13			1.68210			0.841052	+	0.540955i	$0.181937\pi$
$728$		0			0
$729$	3.69854e12			0.485017
$730$		0			0
$731$	1.51056e12			0.195664
$732$		0			0
$733$		−	8.42072e12i		−	1.07741i	−0.842494	−	0.538706i	$-0.818913\pi$
$733$		−	8.42072e12i		−	1.07741i	0.842494	−	0.538706i	$-0.181087\pi$
$734$		0			0
$735$			1.08873e13i			1.37602i
$736$		0			0
$737$	4.75226e11			0.0593330
$738$		0			0
$739$			3.11977e12i			0.384789i	0.981318	+	0.192395i	$0.0616254\pi$
$739$			3.11977e12i			0.384789i	−0.981318	+	0.192395i	$0.938375\pi$
$740$		0			0
$741$	−3.60695e12	+	2.12709e12i	−0.439499	+	0.259182i
$742$		0			0
$743$		−	5.94991e12i		−	0.716244i	−0.933675	−	0.358122i	$-0.883417\pi$
$743$		−	5.94991e12i		−	0.716244i	0.933675	−	0.358122i	$-0.116583\pi$
$744$		0			0
$745$	−5.53054e12			−0.657756
$746$		0			0
$747$			2.19504e12i			0.257929i
$748$		0			0
$749$			2.27249e12i			0.263835i
$750$		0			0
$751$	1.27601e13			1.46377			0.731887	−	0.681426i	$-0.238640\pi$
$751$	1.27601e13			1.46377			0.731887	+	0.681426i	$0.238640\pi$
$752$		0			0
$753$	−1.86813e12			−0.211753
$754$		0			0
$755$	7.64163e12			0.855904
$756$		0			0
$757$	−7.55415e12			−0.836092			−0.418046	−	0.908426i	$-0.637285\pi$
$757$	−7.55415e12			−0.836092			−0.418046	+	0.908426i	$0.637285\pi$
$758$		0			0
$759$			3.48040e12i			0.380663i
$760$		0			0
$761$		−	1.08872e13i		−	1.17675i	−0.808589	−	0.588374i	$-0.799768\pi$
$761$		−	1.08872e13i		−	1.17675i	0.808589	−	0.588374i	$-0.200232\pi$
$762$		0			0
$763$	2.89301e11			0.0309022
$764$		0			0
$765$		−	3.59033e12i		−	0.379016i
$766$		0			0
$767$	−4.16979e12	+	2.45901e12i	−0.435046	+	0.256556i
$768$		0			0
$769$			7.27989e12i			0.750683i	0.926887	+	0.375341i	$0.122475\pi$
$769$			7.27989e12i			0.750683i	−0.926887	+	0.375341i	$0.877525\pi$
$770$		0			0
$771$	−1.90069e11			−0.0193717
$772$		0			0
$773$		−	1.14133e13i		−	1.14975i	−0.818241	−	0.574875i	$-0.805051\pi$
$773$		−	1.14133e13i		−	1.14975i	0.818241	−	0.574875i	$-0.194949\pi$
$774$		0			0
$775$		−	4.44118e12i		−	0.442222i
$776$		0			0
$777$	1.19912e12			0.118023
$778$		0			0
$779$	−5.69362e12			−0.553949
$780$		0			0
$781$	2.33808e12			0.224869
$782$		0			0
$783$	−6.10660e12			−0.580592
$784$		0			0
$785$		−	1.25548e13i		−	1.18004i
$786$		0			0
$787$		−	2.63320e12i		−	0.244679i	−0.992488	−	0.122340i	$-0.960960\pi$
$787$		−	2.63320e12i		−	0.244679i	0.992488	−	0.122340i	$-0.0390397\pi$
$788$		0			0
$789$	−6.78479e11			−0.0623290
$790$		0			0
$791$			2.85730e12i			0.259515i
$792$		0			0
$793$	2.31275e12	+	3.92176e12i	0.207682	+	0.352170i
$794$		0			0
$795$		−	1.96999e13i		−	1.74909i
$796$		0			0
$797$	3.13283e11			0.0275026			0.0137513	−	0.999905i	$-0.495623\pi$
$797$	3.13283e11			0.0275026			0.0137513	+	0.999905i	$0.495623\pi$
$798$		0			0
$799$			1.22069e13i			1.05960i
$800$		0			0
$801$		−	4.23970e12i		−	0.363906i
$802$		0			0
$803$	−1.82718e12			−0.155082
$804$		0			0
$805$	4.85213e12			0.407240
$806$		0			0
$807$	−8.37114e12			−0.694790
$808$		0			0
$809$	7.10201e12			0.582925			0.291463	−	0.956582i	$-0.405858\pi$
$809$	7.10201e12			0.582925			0.291463	+	0.956582i	$0.405858\pi$
$810$		0			0
$811$		−	1.43206e13i		−	1.16243i	−0.813748	−	0.581217i	$-0.802577\pi$
$811$		−	1.43206e13i		−	1.16243i	0.813748	−	0.581217i	$-0.197423\pi$
$812$		0			0
$813$		−	1.99804e13i		−	1.60397i
$814$		0			0
$815$	−2.12610e13			−1.68801
$816$		0			0
$817$			1.24603e12i			0.0978428i
$818$		0			0
$819$	1.02437e12	−	6.04094e11i	0.0795574	−	0.0469166i
$820$		0			0
$821$			1.45899e13i			1.12075i	0.828240	+	0.560373i	$0.189342\pi$
$821$			1.45899e13i			1.12075i	−0.828240	+	0.560373i	$0.810658\pi$
$822$		0			0
$823$	2.10512e13			1.59947			0.799736	−	0.600351i	$-0.204973\pi$
$823$	2.10512e13			1.59947			0.799736	+	0.600351i	$0.204973\pi$
$824$		0			0
$825$		−	3.00659e12i		−	0.225960i
$826$		0			0
$827$		−	1.86271e13i		−	1.38474i	−0.721541	−	0.692372i	$-0.756566\pi$
$827$		−	1.86271e13i		−	1.38474i	0.721541	−	0.692372i	$-0.243434\pi$
$828$		0			0
$829$	4.93039e12			0.362565			0.181283	−	0.983431i	$-0.441975\pi$
$829$	4.93039e12			0.362565			0.181283	+	0.983431i	$0.441975\pi$
$830$		0			0
$831$	−2.13472e13			−1.55287
$832$		0			0
$833$	1.13342e13			0.815619
$834$		0			0
$835$	−1.62704e13			−1.15827
$836$		0			0
$837$			7.21522e12i			0.508142i
$838$		0			0
$839$		−	2.14228e13i		−	1.49261i	−0.665603	−	0.746306i	$-0.731825\pi$
$839$		−	2.14228e13i		−	1.49261i	0.665603	−	0.746306i	$-0.268175\pi$
$840$		0			0
$841$	−6.28769e12			−0.433420
$842$		0			0
$843$		−	1.51155e13i		−	1.03085i
$844$		0			0
$845$	1.67664e13	+	9.27177e12i	1.13132	+	0.625615i
$846$		0			0
$847$			3.81744e12i			0.254857i
$848$		0			0
$849$	2.00159e11			0.0132218
$850$		0			0
$851$			6.35266e12i			0.415214i
$852$		0			0
$853$			2.03744e13i			1.31769i	0.752278	+	0.658846i	$0.228955\pi$
$853$			2.03744e13i			1.31769i	−0.752278	+	0.658846i	$0.771045\pi$
$854$		0			0
$855$	2.96158e12			0.189529
$856$		0			0
$857$	2.22501e13			1.40902			0.704512	−	0.709692i	$-0.251166\pi$
$857$	2.22501e13			1.40902			0.704512	+	0.709692i	$0.251166\pi$
$858$		0			0
$859$	1.78122e13			1.11622			0.558108	−	0.829768i	$-0.311527\pi$
$859$	1.78122e13			1.11622			0.558108	+	0.829768i	$0.311527\pi$
$860$		0			0
$861$	6.49383e12			0.402705
$862$		0			0
$863$		−	1.59283e12i		−	0.0977511i	−0.998805	−	0.0488755i	$-0.984436\pi$
$863$		−	1.59283e12i		−	0.0977511i	0.998805	−	0.0488755i	$-0.0155638\pi$
$864$		0			0
$865$		−	2.85494e13i		−	1.73390i
$866$		0			0
$867$	4.18782e12			0.251711
$868$		0			0
$869$		−	7.68054e12i		−	0.456881i
$870$		0			0
$871$	2.97589e12	−	1.75494e12i	0.175200	−	0.103319i
$872$		0			0
$873$		−	5.71353e11i		−	0.0332920i
$874$		0			0
$875$	2.05268e12			0.118382
$876$		0			0
$877$			2.45278e13i			1.40010i	0.714092	+	0.700052i	$0.246840\pi$
$877$			2.45278e13i			1.40010i	−0.714092	+	0.700052i	$0.753160\pi$
$878$		0			0
$879$		−	2.89270e13i		−	1.63438i
$880$		0			0
$881$	7.05046e12			0.394299			0.197150	−	0.980373i	$-0.436832\pi$
$881$	7.05046e12			0.394299			0.197150	+	0.980373i	$0.436832\pi$
$882$		0			0
$883$	7.94625e12			0.439885			0.219942	−	0.975513i	$-0.429413\pi$
$883$	7.94625e12			0.439885			0.219942	+	0.975513i	$0.429413\pi$
$884$		0			0
$885$	1.37497e13			0.753438
$886$		0			0
$887$	−2.35837e13			−1.27925			−0.639626	−	0.768686i	$-0.720911\pi$
$887$	−2.35837e13			−1.27925			−0.639626	+	0.768686i	$0.720911\pi$
$888$		0			0
$889$			5.03868e12i			0.270557i
$890$		0			0
$891$			6.70413e12i			0.356363i
$892$		0			0
$893$	−1.00692e13			−0.529861
$894$		0			0
$895$			4.18616e13i			2.18078i
$896$		0			0
$897$	1.28526e13	+	2.17944e13i	0.662867	+	1.12404i
$898$		0			0
$899$		−	9.71165e12i		−	0.495878i
$900$		0			0
$901$	−2.05085e13			−1.03675
$902$		0			0
$903$		−	1.42115e12i		−	0.0711289i
$904$		0			0
$905$		−	1.56909e13i		−	0.777553i
$906$		0			0
$907$	1.43910e13			0.706087			0.353043	−	0.935607i	$-0.385147\pi$
$907$	1.43910e13			0.706087			0.353043	+	0.935607i	$0.385147\pi$
$908$		0			0
$909$	8.92796e12			0.433725
$910$		0			0
$911$	1.35452e13			0.651560			0.325780	−	0.945446i	$-0.394373\pi$
$911$	1.35452e13			0.651560			0.325780	+	0.945446i	$0.394373\pi$
$912$		0			0
$913$	4.76435e12			0.226926
$914$		0			0
$915$		−	1.29318e13i		−	0.609908i
$916$		0			0
$917$		−	8.34465e12i		−	0.389714i
$918$		0			0
$919$	−1.25139e13			−0.578728			−0.289364	−	0.957219i	$-0.593444\pi$
$919$	−1.25139e13			−0.578728			−0.289364	+	0.957219i	$0.593444\pi$
$920$		0			0
$921$		−	3.58359e13i		−	1.64116i
$922$		0			0
$923$	1.46412e13	−	8.63420e12i	0.664000	−	0.391575i
$924$		0			0
$925$		−	5.48782e12i		−	0.246469i
$926$		0			0
$927$	4.41637e12			0.196429
$928$		0			0
$929$		−	2.61952e12i		−	0.115386i	−0.998334	−	0.0576928i	$-0.981626\pi$
$929$		−	2.61952e12i		−	0.115386i	0.998334	−	0.0576928i	$-0.0183744\pi$
$930$		0			0
$931$			9.34931e12i			0.407855i
$932$		0			0
$933$	−1.20811e12			−0.0521962
$934$		0			0
$935$	−7.79283e12			−0.333459
$936$		0			0
$937$	−1.12398e11			−0.00476355			−0.00238178	−	0.999997i	$-0.500758\pi$
$937$	−1.12398e11			−0.00476355			−0.00238178	+	0.999997i	$0.500758\pi$
$938$		0			0
$939$	−1.04382e13			−0.438158
$940$		0			0
$941$		−	1.88869e12i		−	0.0785247i	−0.999229	−	0.0392623i	$-0.987499\pi$
$941$		−	1.88869e12i		−	0.0785247i	0.999229	−	0.0392623i	$-0.0125008\pi$
$942$		0			0
$943$			3.44028e13i			1.41674i
$944$		0			0
$945$	6.80970e12			0.277770
$946$		0			0
$947$			3.94169e13i			1.59260i	0.604901	+	0.796301i	$0.293213\pi$
$947$			3.94169e13i			1.59260i	−0.604901	+	0.796301i	$0.706787\pi$
$948$		0			0
$949$	−1.14419e13	+	6.74751e12i	−0.457930	+	0.270051i
$950$		0			0
$951$		−	4.60339e13i		−	1.82501i
$952$		0			0
$953$	−1.99699e13			−0.784256			−0.392128	−	0.919911i	$-0.628261\pi$
$953$	−1.99699e13			−0.784256			−0.392128	+	0.919911i	$0.628261\pi$
$954$		0			0
$955$			2.69600e13i			1.04883i
$956$		0			0
$957$		−	6.57459e12i		−	0.253376i
$958$		0			0
$959$	7.60859e12			0.290483
$960$		0			0
$961$	1.49649e13			0.566002
$962$		0			0
$963$	8.38105e12			0.314037
$964$		0			0
$965$	−2.33409e13			−0.866453
$966$		0			0
$967$		−	1.37561e13i		−	0.505912i	−0.967478	−	0.252956i	$-0.918597\pi$
$967$		−	1.37561e13i		−	0.505912i	0.967478	−	0.252956i	$-0.0814029\pi$
$968$		0			0
$969$		−	1.23820e13i		−	0.451162i
$970$		0			0
$971$	4.83327e13			1.74484			0.872418	−	0.488760i	$-0.162551\pi$
$971$	4.83327e13			1.74484			0.872418	+	0.488760i	$0.162551\pi$
$972$		0			0
$973$		−	5.77709e12i		−	0.206634i
$974$		0			0
$975$	−1.11029e13	−	1.88274e13i	−0.393474	−	0.667221i
$976$		0			0
$977$			2.58790e13i			0.908703i	0.890823	+	0.454351i	$0.150129\pi$
$977$			2.58790e13i			0.908703i	−0.890823	+	0.454351i	$0.849871\pi$
$978$		0			0
$979$	−9.20230e12			−0.320165
$980$		0			0
$981$		−	1.06696e12i		−	0.0367821i
$982$		0			0
$983$		−	2.57154e13i		−	0.878422i	−0.898384	−	0.439211i	$-0.855258\pi$
$983$		−	2.57154e13i		−	0.878422i	0.898384	−	0.439211i	$-0.144742\pi$
$984$		0			0
$985$	−2.60699e13			−0.882423
$986$		0			0
$987$	1.14843e13			0.385193
$988$		0			0
$989$	7.52895e12			0.250237
$990$		0			0
$991$	6.55661e12			0.215947			0.107974	−	0.994154i	$-0.465564\pi$
$991$	6.55661e12			0.215947			0.107974	+	0.994154i	$0.465564\pi$
$992$		0			0
$993$			2.06409e13i			0.673685i
$994$		0			0
$995$			3.29578e13i			1.06599i
$996$		0			0
$997$	−1.47827e13			−0.473834			−0.236917	−	0.971530i	$-0.576137\pi$
$997$	−1.47827e13			−0.473834			−0.236917	+	0.971530i	$0.576137\pi$
$998$		0			0
$999$			8.91561e12i			0.283209i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.f.d.129.8		32
4.3	odd	2		104.10.f.a.25.25	✓	32
13.12	even	2	inner	208.10.f.d.129.7		32
52.51	odd	2		104.10.f.a.25.26	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.25	✓	32	4.3	odd	2
104.10.f.a.25.26	yes	32	52.51	odd	2
208.10.f.d.129.7		32	13.12	even	2	inner
208.10.f.d.129.8		32	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-161.892 q^{3} -1806.71i q^{5} +1769.55i q^{7} +6526.18 q^{9} -14165.1i q^{11} +(-52309.9 - 88702.7i) q^{13} +292493. i q^{15} +304500. q^{17} +251175. i q^{19} -286476. i q^{21} +1.51769e6 q^{23} -1.31107e6 q^{25} +2.12999e6 q^{27} -2.86696e6 q^{29} +3.38744e6i q^{31} +2.29323e6i q^{33} +3.19706e6 q^{35} +4.18575e6i q^{37} +(8.46857e6 + 1.43603e7i) q^{39} +2.26679e7i q^{41} +4.96081e6 q^{43} -1.17909e7i q^{45} +4.00883e7i q^{47} +3.72223e7 q^{49} -4.92962e7 q^{51} -6.73516e7 q^{53} -2.55923e7 q^{55} -4.06633e7i q^{57} -4.70086e7i q^{59} -4.42124e7 q^{61} +1.15484e7i q^{63} +(-1.60260e8 + 9.45087e7i) q^{65} +3.35490e7i q^{67} -2.45702e8 q^{69} +1.65059e8i q^{71} -1.28991e8i q^{73} +2.12253e8 q^{75} +2.50658e7 q^{77} +5.42215e8 q^{79} -4.73284e8 q^{81} +3.36344e8i q^{83} -5.50142e8i q^{85} +4.64139e8 q^{87} -6.49645e8i q^{89} +(1.56964e8 - 9.25647e7i) q^{91} -5.48401e8i q^{93} +4.53800e8 q^{95} -8.75479e7i q^{97} -9.24441e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) \) 32 * q + 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 - 3171556 * q^23 - 13526722 * q^25 + 3694974 * q^27 + 8833508 * q^29 + 8281126 * q^35 + 12056860 * q^39 - 89959038 * q^43 - 172344874 * q^49 - 51206562 * q^51 + 139521112 * q^53 + 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 + 604473728 * q^75 - 1666197144 * q^77 - 1165982916 * q^79 + 859636544 * q^81 + 1738691296 * q^87 + 291718146 * q^91 - 1741143356 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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