LMFDB - Embedded newform 208.10.f.d.129.9

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			129.9
Character	$\chi$	$=$	208.129
Dual form			208.10.f.d.129.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-132.434 q^{3} -362.362i q^{5} -3483.24i q^{7} -2144.23 q^{9} -12306.6i q^{11} +(100932. - 20425.6i) q^{13} +47989.0i q^{15} -482437. q^{17} -791929. i q^{19} +461299. i q^{21} +263044. q^{23} +1.82182e6 q^{25} +2.89067e6 q^{27} -22202.9 q^{29} -6.64793e6i q^{31} +1.62981e6i q^{33} -1.26219e6 q^{35} +1.71134e7i q^{37} +(-1.33668e7 + 2.70504e6i) q^{39} -1.77323e7i q^{41} +2.51637e7 q^{43} +776987. i q^{45} -1.23894e6i q^{47} +2.82207e7 q^{49} +6.38910e7 q^{51} +1.02344e7 q^{53} -4.45943e6 q^{55} +1.04878e8i q^{57} +4.48760e7i q^{59} +4.65679e7 q^{61} +7.46886e6i q^{63} +(-7.40144e6 - 3.65739e7i) q^{65} -8.12489e7i q^{67} -3.48360e7 q^{69} -4.17437e8i q^{71} +4.28566e8i q^{73} -2.41271e8 q^{75} -4.28667e7 q^{77} +4.90206e8 q^{79} -3.40618e8 q^{81} -6.62047e8i q^{83} +1.74817e8i q^{85} +2.94041e6 q^{87} +3.17115e8i q^{89} +(-7.11470e7 - 3.51570e8i) q^{91} +8.80412e8i q^{93} -2.86964e8 q^{95} +1.06090e8i q^{97} +2.63881e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−132.434			−0.943961			−0.471980	−	0.881609i	$-0.656461\pi$
$3$	−132.434			−0.943961			−0.471980	+	0.881609i	$0.656461\pi$
$4$		0			0
$5$		−	362.362i		−	0.259285i	−0.991561	−	0.129642i	$-0.958617\pi$
$5$		−	362.362i		−	0.259285i	0.991561	−	0.129642i	$-0.0413829\pi$
$6$		0			0
$7$		−	3483.24i		−	0.548330i	−0.961683	−	0.274165i	$-0.911599\pi$
$7$		−	3483.24i		−	0.548330i	0.961683	−	0.274165i	$-0.0884014\pi$
$8$		0			0
$9$	−2144.23			−0.108938
$10$		0			0
$11$		−	12306.6i		−	0.253437i	−0.991939	−	0.126718i	$-0.959556\pi$
$11$		−	12306.6i		−	0.253437i	0.991939	−	0.126718i	$-0.0404445\pi$
$12$		0			0
$13$	100932.	−	20425.6i	0.980132	−	0.198348i
$13$	100932.	−	20425.6i	0.980132	−	0.198348i
$14$		0			0
$15$			47989.0i			0.244755i
$16$		0			0
$17$	−482437.			−1.40094			−0.700471	−	0.713681i	$-0.747027\pi$
$17$	−482437.			−1.40094			−0.700471	+	0.713681i	$0.747027\pi$
$18$		0			0
$19$		−	791929.i		−	1.39410i	−0.717021	−	0.697051i	$-0.754495\pi$
$19$		−	791929.i		−	1.39410i	0.717021	−	0.697051i	$-0.245505\pi$
$20$		0			0
$21$			461299.i			0.517602i
$22$		0			0
$23$	263044.			0.195999			0.0979993	−	0.995186i	$-0.468756\pi$
$23$	263044.			0.195999			0.0979993	+	0.995186i	$0.468756\pi$
$24$		0			0
$25$	1.82182e6			0.932771
$26$		0			0
$27$	2.89067e6			1.04679
$28$		0			0
$29$	−22202.9			−0.00582932			−0.00291466	−	0.999996i	$-0.500928\pi$
$29$	−22202.9			−0.00582932			−0.00291466	+	0.999996i	$0.500928\pi$
$30$		0			0
$31$		−	6.64793e6i		−	1.29288i	−0.762964	−	0.646441i	$-0.776257\pi$
$31$		−	6.64793e6i		−	1.29288i	0.762964	−	0.646441i	$-0.223743\pi$
$32$		0			0
$33$			1.62981e6i			0.239235i
$34$		0			0
$35$	−1.26219e6			−0.142174
$36$		0			0
$37$			1.71134e7i			1.50117i	0.660776	+	0.750583i	$0.270227\pi$
$37$			1.71134e7i			1.50117i	−0.660776	+	0.750583i	$0.729773\pi$
$38$		0			0
$39$	−1.33668e7	+	2.70504e6i	−0.925206	+	0.187233i
$40$		0			0
$41$		−	1.77323e7i		−	0.980025i	−0.871715	−	0.490013i	$-0.836992\pi$
$41$		−	1.77323e7i		−	0.980025i	0.871715	−	0.490013i	$-0.163008\pi$
$42$		0			0
$43$	2.51637e7			1.12245			0.561224	−	0.827664i	$-0.310331\pi$
$43$	2.51637e7			1.12245			0.561224	+	0.827664i	$0.310331\pi$
$44$		0			0
$45$			776987.i			0.0282460i
$46$		0			0
$47$		−	1.23894e6i		−	0.0370348i	−0.999829	−	0.0185174i	$-0.994105\pi$
$47$		−	1.23894e6i		−	0.0370348i	0.999829	−	0.0185174i	$-0.00589462\pi$
$48$		0			0
$49$	2.82207e7			0.699335
$50$		0			0
$51$	6.38910e7			1.32243
$52$		0			0
$53$	1.02344e7			0.178164			0.0890821	−	0.996024i	$-0.471607\pi$
$53$	1.02344e7			0.178164			0.0890821	+	0.996024i	$0.471607\pi$
$54$		0			0
$55$	−4.45943e6			−0.0657124
$56$		0			0
$57$			1.04878e8i			1.31598i
$58$		0			0
$59$			4.48760e7i			0.482147i	0.970507	+	0.241074i	$0.0774995\pi$
$59$			4.48760e7i			0.482147i	−0.970507	+	0.241074i	$0.922500\pi$
$60$		0			0
$61$	4.65679e7			0.430629			0.215314	−	0.976545i	$-0.430922\pi$
$61$	4.65679e7			0.430629			0.215314	+	0.976545i	$0.430922\pi$
$62$		0			0
$63$			7.46886e6i			0.0597341i
$64$		0			0
$65$	−7.40144e6	−	3.65739e7i	−0.0514287	−	0.254133i
$66$		0			0
$67$		−	8.12489e7i		−	0.492585i	−0.969196	−	0.246292i	$-0.920788\pi$
$67$		−	8.12489e7i		−	0.492585i	0.969196	−	0.246292i	$-0.0792123\pi$
$68$		0			0
$69$	−3.48360e7			−0.185015
$70$		0			0
$71$		−	4.17437e8i		−	1.94952i	−0.223249	−	0.974761i	$-0.571666\pi$
$71$		−	4.17437e8i		−	1.94952i	0.223249	−	0.974761i	$-0.428334\pi$
$72$		0			0
$73$			4.28566e8i			1.76630i	0.469090	+	0.883150i	$0.344582\pi$
$73$			4.28566e8i			1.76630i	−0.469090	+	0.883150i	$0.655418\pi$
$74$		0			0
$75$	−2.41271e8			−0.880499
$76$		0			0
$77$	−4.28667e7			−0.138967
$78$		0			0
$79$	4.90206e8			1.41598			0.707989	−	0.706223i	$-0.249602\pi$
$79$	4.90206e8			1.41598			0.707989	+	0.706223i	$0.249602\pi$
$80$		0			0
$81$	−3.40618e8			−0.879194
$82$		0			0
$83$		−	6.62047e8i		−	1.53122i	−0.643305	−	0.765610i	$-0.722437\pi$
$83$		−	6.62047e8i		−	1.53122i	0.643305	−	0.765610i	$-0.277563\pi$
$84$		0			0
$85$			1.74817e8i			0.363243i
$86$		0			0
$87$	2.94041e6			0.00550265
$88$		0			0
$89$			3.17115e8i			0.535749i	0.963454	+	0.267875i	$0.0863213\pi$
$89$			3.17115e8i			0.535749i	−0.963454	+	0.267875i	$0.913679\pi$
$90$		0			0
$91$	−7.11470e7	−	3.51570e8i	−0.108760	−	0.537435i
$92$		0			0
$93$			8.80412e8i			1.22043i
$94$		0			0
$95$	−2.86964e8			−0.361470
$96$		0			0
$97$			1.06090e8i			0.121676i	0.998148	+	0.0608378i	$0.0193772\pi$
$97$			1.06090e8i			0.121676i	−0.998148	+	0.0608378i	$0.980623\pi$
$98$		0			0
$99$			2.63881e7i			0.0276090i
$100$		0			0
$101$	−1.98728e9			−1.90026			−0.950130	−	0.311854i	$-0.899050\pi$
$101$	−1.98728e9			−1.90026			−0.950130	+	0.311854i	$0.899050\pi$
$102$		0			0
$103$	−1.35702e9			−1.18801			−0.594004	−	0.804462i	$-0.702454\pi$
$103$	−1.35702e9			−1.18801			−0.594004	+	0.804462i	$0.702454\pi$
$104$		0			0
$105$	1.67157e8			0.134206
$106$		0			0
$107$	−1.41925e9			−1.04672			−0.523361	−	0.852111i	$-0.675322\pi$
$107$	−1.41925e9			−1.04672			−0.523361	+	0.852111i	$0.675322\pi$
$108$		0			0
$109$		−	1.84329e9i		−	1.25076i	−0.780321	−	0.625379i	$-0.784944\pi$
$109$		−	1.84329e9i		−	1.25076i	0.780321	−	0.625379i	$-0.215056\pi$
$110$		0			0
$111$		−	2.26640e9i		−	1.41704i
$112$		0			0
$113$	−2.02691e9			−1.16945			−0.584725	−	0.811231i	$-0.698798\pi$
$113$	−2.02691e9			−1.16945			−0.584725	+	0.811231i	$0.698798\pi$
$114$		0			0
$115$		−	9.53170e7i		−	0.0508195i
$116$		0			0
$117$	−2.16422e8	+	4.37971e7i	−0.106774	+	0.0216077i
$118$		0			0
$119$			1.68044e9i			0.768178i
$120$		0			0
$121$	2.20650e9			0.935770
$122$		0			0
$123$			2.34836e9i			0.925105i
$124$		0			0
$125$		−	1.36789e9i		−	0.501138i
$126$		0			0
$127$	−3.24504e9			−1.10689			−0.553443	−	0.832887i	$-0.686686\pi$
$127$	−3.24504e9			−1.10689			−0.553443	+	0.832887i	$0.686686\pi$
$128$		0			0
$129$	−3.33253e9			−1.05955
$130$		0			0
$131$	1.34206e8			0.0398155			0.0199078	−	0.999802i	$-0.493663\pi$
$131$	1.34206e8			0.0398155			0.0199078	+	0.999802i	$0.493663\pi$
$132$		0			0
$133$	−2.75847e9			−0.764428
$134$		0			0
$135$		−	1.04747e9i		−	0.271418i
$136$		0			0
$137$		−	1.49747e9i		−	0.363175i	−0.983375	−	0.181588i	$-0.941876\pi$
$137$		−	1.49747e9i		−	0.363175i	0.983375	−	0.181588i	$-0.0581236\pi$
$138$		0			0
$139$	−1.38616e9			−0.314955			−0.157477	−	0.987523i	$-0.550336\pi$
$139$	−1.38616e9			−0.314955			−0.157477	+	0.987523i	$0.550336\pi$
$140$		0			0
$141$			1.64078e8i			0.0349594i
$142$		0			0
$143$	−2.51369e8	−	1.24213e9i	−0.0502688	−	0.248402i
$144$		0			0
$145$			8.04547e6i			0.00151145i
$146$		0			0
$147$	−3.73738e9			−0.660144
$148$		0			0
$149$		−	4.28758e9i		−	0.712647i	−0.934363	−	0.356323i	$-0.884030\pi$
$149$		−	4.28758e9i		−	0.712647i	0.934363	−	0.356323i	$-0.115970\pi$
$150$		0			0
$151$			2.17856e9i			0.341014i	0.985356	+	0.170507i	$0.0545406\pi$
$151$			2.17856e9i			0.341014i	−0.985356	+	0.170507i	$0.945459\pi$
$152$		0			0
$153$	1.03446e9			0.152616
$154$		0			0
$155$	−2.40895e9			−0.335225
$156$		0			0
$157$	−7.65180e9			−1.00511			−0.502557	−	0.864544i	$-0.667607\pi$
$157$	−7.65180e9			−1.00511			−0.502557	+	0.864544i	$0.667607\pi$
$158$		0			0
$159$	−1.35538e9			−0.168180
$160$		0			0
$161$		−	9.16244e8i		−	0.107472i
$162$		0			0
$163$			5.36532e9i			0.595322i	0.954672	+	0.297661i	$0.0962064\pi$
$163$			5.36532e9i			0.595322i	−0.954672	+	0.297661i	$0.903794\pi$
$164$		0			0
$165$	5.90580e8			0.0620299
$166$		0			0
$167$			1.27058e10i			1.26409i	0.774930	+	0.632047i	$0.217785\pi$
$167$			1.27058e10i			1.26409i	−0.774930	+	0.632047i	$0.782215\pi$
$168$		0			0
$169$	9.77009e9	−	4.12319e9i	0.921316	−	0.388815i
$170$		0			0
$171$			1.69808e9i			0.151871i
$172$		0			0
$173$	−4.65892e9			−0.395438			−0.197719	−	0.980259i	$-0.563353\pi$
$173$	−4.65892e9			−0.395438			−0.197719	+	0.980259i	$0.563353\pi$
$174$		0			0
$175$		−	6.34583e9i		−	0.511466i
$176$		0			0
$177$		−	5.94310e9i		−	0.455128i
$178$		0			0
$179$	−9.51770e9			−0.692936			−0.346468	−	0.938062i	$-0.612619\pi$
$179$	−9.51770e9			−0.692936			−0.346468	+	0.938062i	$0.612619\pi$
$180$		0			0
$181$	−1.93702e10			−1.34147			−0.670735	−	0.741697i	$-0.734021\pi$
$181$	−1.93702e10			−1.34147			−0.670735	+	0.741697i	$0.734021\pi$
$182$		0			0
$183$	−6.16718e9			−0.406496
$184$		0			0
$185$	6.20124e9			0.389230
$186$		0			0
$187$			5.93714e9i			0.355051i
$188$		0			0
$189$		−	1.00689e10i		−	0.573988i
$190$		0			0
$191$	−2.72271e9			−0.148030			−0.0740152	−	0.997257i	$-0.523581\pi$
$191$	−2.72271e9			−0.148030			−0.0740152	+	0.997257i	$0.523581\pi$
$192$		0			0
$193$		−	2.81939e10i		−	1.46267i	−0.682016	−	0.731337i	$-0.738896\pi$
$193$		−	2.81939e10i		−	1.46267i	0.682016	−	0.731337i	$-0.261104\pi$
$194$		0			0
$195$	9.80202e8	+	4.84363e9i	0.0485467	+	0.239892i
$196$		0			0
$197$		−	1.88074e10i		−	0.889672i	−0.895612	−	0.444836i	$-0.853262\pi$
$197$		−	1.88074e10i		−	0.889672i	0.895612	−	0.444836i	$-0.146738\pi$
$198$		0			0
$199$	−1.68577e10			−0.762009			−0.381004	−	0.924573i	$-0.624422\pi$
$199$	−1.68577e10			−0.762009			−0.381004	+	0.924573i	$0.624422\pi$
$200$		0			0
$201$			1.07601e10i			0.464980i
$202$		0			0
$203$			7.73378e7i			0.00319639i
$204$		0			0
$205$	−6.42550e9			−0.254106
$206$		0			0
$207$	−5.64027e8			−0.0213517
$208$		0			0
$209$	−9.74593e9			−0.353317
$210$		0			0
$211$	−2.44393e10			−0.848826			−0.424413	−	0.905469i	$-0.639519\pi$
$211$	−2.44393e10			−0.848826			−0.424413	+	0.905469i	$0.639519\pi$
$212$		0			0
$213$			5.52829e10i			1.84027i
$214$		0			0
$215$		−	9.11835e9i		−	0.291034i
$216$		0			0
$217$	−2.31563e10			−0.708926
$218$		0			0
$219$		−	5.67567e10i		−	1.66732i
$220$		0			0
$221$	−4.86934e10	+	9.85404e9i	−1.37311	+	0.277875i
$222$		0			0
$223$			1.52796e10i			0.413753i	0.978367	+	0.206876i	$0.0663298\pi$
$223$			1.52796e10i			0.413753i	−0.978367	+	0.206876i	$0.933670\pi$
$224$		0			0
$225$	−3.90640e9			−0.101614
$226$		0			0
$227$		−	9.89806e9i		−	0.247419i	−0.992318	−	0.123710i	$-0.960521\pi$
$227$		−	9.89806e9i		−	0.247419i	0.992318	−	0.123710i	$-0.0394792\pi$
$228$		0			0
$229$		−	1.61656e10i		−	0.388447i	−0.980957	−	0.194224i	$-0.937781\pi$
$229$		−	1.61656e10i		−	0.388447i	0.980957	−	0.194224i	$-0.0622188\pi$
$230$		0			0
$231$	5.67701e9			0.131179
$232$		0			0
$233$	5.56739e10			1.23751			0.618757	−	0.785583i	$-0.287637\pi$
$233$	5.56739e10			1.23751			0.618757	+	0.785583i	$0.287637\pi$
$234$		0			0
$235$	−4.48945e8			−0.00960257
$236$		0			0
$237$	−6.49199e10			−1.33663
$238$		0			0
$239$		−	2.46828e10i		−	0.489332i	−0.969607	−	0.244666i	$-0.921322\pi$
$239$		−	2.46828e10i		−	0.489332i	0.969607	−	0.244666i	$-0.0786783\pi$
$240$		0			0
$241$			9.43996e10i			1.80258i	0.433221	+	0.901288i	$0.357377\pi$
$241$			9.43996e10i			1.80258i	−0.433221	+	0.901288i	$0.642623\pi$
$242$		0			0
$243$	−1.17876e10			−0.216869
$244$		0			0
$245$		−	1.02261e10i		−	0.181327i
$246$		0			0
$247$	−1.61756e10	−	7.99310e10i	−0.276518	−	1.36640i
$248$		0			0
$249$			8.76776e10i			1.44541i
$250$		0			0
$251$	−4.32735e10			−0.688161			−0.344081	−	0.938940i	$-0.611809\pi$
$251$	−4.32735e10			−0.688161			−0.344081	+	0.938940i	$0.611809\pi$
$252$		0			0
$253$		−	3.23717e9i		−	0.0496733i
$254$		0			0
$255$		−	2.31517e10i		−	0.342887i
$256$		0			0
$257$	3.53120e10			0.504921			0.252461	−	0.967607i	$-0.418760\pi$
$257$	3.53120e10			0.504921			0.252461	+	0.967607i	$0.418760\pi$
$258$		0			0
$259$	5.96100e10			0.823134
$260$		0			0
$261$	4.76081e7			0.000635036
$262$		0			0
$263$	9.13037e10			1.17676			0.588380	−	0.808585i	$-0.299766\pi$
$263$	9.13037e10			1.17676			0.588380	+	0.808585i	$0.299766\pi$
$264$		0			0
$265$		−	3.70855e9i		−	0.0461953i
$266$		0			0
$267$		−	4.19968e10i		−	0.505726i
$268$		0			0
$269$	2.84278e10			0.331023			0.165512	−	0.986208i	$-0.447072\pi$
$269$	2.84278e10			0.331023			0.165512	+	0.986208i	$0.447072\pi$
$270$		0			0
$271$			1.26666e11i			1.42658i	0.700868	+	0.713291i	$0.252796\pi$
$271$			1.26666e11i			1.42658i	−0.700868	+	0.713291i	$0.747204\pi$
$272$		0			0
$273$	9.42229e9	+	4.65599e10i	0.102665	+	0.507318i
$274$		0			0
$275$		−	2.24203e10i		−	0.236399i
$276$		0			0
$277$	3.70737e8			0.00378362			0.00189181	−	0.999998i	$-0.499398\pi$
$277$	3.70737e8			0.00378362			0.00189181	+	0.999998i	$0.499398\pi$
$278$		0			0
$279$			1.42547e10i			0.140844i
$280$		0			0
$281$		−	1.18871e11i		−	1.13736i	−0.822560	−	0.568678i	$-0.807455\pi$
$281$		−	1.18871e11i		−	1.13736i	0.822560	−	0.568678i	$-0.192545\pi$
$282$		0			0
$283$	1.15032e11			1.06606			0.533029	−	0.846097i	$-0.321053\pi$
$283$	1.15032e11			1.06606			0.533029	+	0.846097i	$0.321053\pi$
$284$		0			0
$285$	3.80039e10			0.341213
$286$		0			0
$287$	−6.17657e10			−0.537377
$288$		0			0
$289$	1.14157e11			0.962638
$290$		0			0
$291$		−	1.40500e10i		−	0.114857i
$292$		0			0
$293$		−	1.61534e10i		−	0.128044i	−0.997948	−	0.0640219i	$-0.979607\pi$
$293$		−	1.61534e10i		−	0.128044i	0.997948	−	0.0640219i	$-0.0203927\pi$
$294$		0			0
$295$	1.62613e10			0.125013
$296$		0			0
$297$		−	3.55742e10i		−	0.265296i
$298$		0			0
$299$	2.65496e10	−	5.37282e9i	0.192104	−	0.0388760i
$300$		0			0
$301$		−	8.76510e10i		−	0.615471i
$302$		0			0
$303$	2.63184e11			1.79377
$304$		0			0
$305$		−	1.68744e10i		−	0.111655i
$306$		0			0
$307$			8.26154e10i			0.530809i	0.964137	+	0.265405i	$0.0855055\pi$
$307$			8.26154e10i			0.530809i	−0.964137	+	0.265405i	$0.914494\pi$
$308$		0			0
$309$	1.79716e11			1.12143
$310$		0			0
$311$	7.43197e10			0.450487			0.225244	−	0.974302i	$-0.427682\pi$
$311$	7.43197e10			0.450487			0.225244	+	0.974302i	$0.427682\pi$
$312$		0			0
$313$	1.36875e11			0.806075			0.403038	−	0.915183i	$-0.367954\pi$
$313$	1.36875e11			0.806075			0.403038	+	0.915183i	$0.367954\pi$
$314$		0			0
$315$	2.70643e9			0.0154881
$316$		0			0
$317$			2.08252e11i			1.15831i	0.815219	+	0.579153i	$0.196616\pi$
$317$			2.08252e11i			1.15831i	−0.815219	+	0.579153i	$0.803384\pi$
$318$		0			0
$319$			2.73241e8i			0.00147737i
$320$		0			0
$321$	1.87957e11			0.988064
$322$		0			0
$323$			3.82055e11i			1.95306i
$324$		0			0
$325$	1.83880e11	−	3.72117e10i	0.914239	−	0.185014i
$326$		0			0
$327$			2.44114e11i			1.18067i
$328$		0			0
$329$	−4.31553e9			−0.0203073
$330$		0			0
$331$		−	1.83671e11i		−	0.841034i	−0.907285	−	0.420517i	$-0.861849\pi$
$331$		−	1.83671e11i		−	0.841034i	0.907285	−	0.420517i	$-0.138151\pi$
$332$		0			0
$333$		−	3.66951e10i		−	0.163534i
$334$		0			0
$335$	−2.94415e10			−0.127720
$336$		0			0
$337$	−2.86391e10			−0.120955			−0.0604776	−	0.998170i	$-0.519262\pi$
$337$	−2.86391e10			−0.120955			−0.0604776	+	0.998170i	$0.519262\pi$
$338$		0			0
$339$	2.68432e11			1.10392
$340$		0			0
$341$	−8.18132e10			−0.327664
$342$		0			0
$343$		−	2.38860e11i		−	0.931796i
$344$		0			0
$345$			1.26232e10i			0.0479716i
$346$		0			0
$347$	−1.39827e11			−0.517736			−0.258868	−	0.965913i	$-0.583349\pi$
$347$	−1.39827e11			−0.517736			−0.258868	+	0.965913i	$0.583349\pi$
$348$		0			0
$349$		−	8.05612e10i		−	0.290678i	−0.989382	−	0.145339i	$-0.953573\pi$
$349$		−	8.05612e10i		−	0.290678i	0.989382	−	0.145339i	$-0.0464272\pi$
$350$		0			0
$351$	2.91761e11	−	5.90435e10i	1.02600	−	0.207630i
$352$		0			0
$353$		−	6.38499e10i		−	0.218864i	−0.993994	−	0.109432i	$-0.965097\pi$
$353$		−	6.38499e10i		−	0.218864i	0.993994	−	0.109432i	$-0.0349031\pi$
$354$		0			0
$355$	−1.51263e11			−0.505482
$356$		0			0
$357$		−	2.22548e11i		−	0.725130i
$358$		0			0
$359$			4.31792e11i			1.37199i	0.727608	+	0.685993i	$0.240632\pi$
$359$			4.31792e11i			1.37199i	−0.727608	+	0.685993i	$0.759368\pi$
$360$		0			0
$361$	−3.04463e11			−0.943522
$362$		0			0
$363$	−2.92215e11			−0.883330
$364$		0			0
$365$	1.55296e11			0.457975
$366$		0			0
$367$	6.40722e10			0.184362			0.0921812	−	0.995742i	$-0.470616\pi$
$367$	6.40722e10			0.184362			0.0921812	+	0.995742i	$0.470616\pi$
$368$		0			0
$369$			3.80221e10i			0.106762i
$370$		0			0
$371$		−	3.56488e10i		−	0.0976928i
$372$		0			0
$373$	−1.38528e11			−0.370552			−0.185276	−	0.982687i	$-0.559318\pi$
$373$	−1.38528e11			−0.370552			−0.185276	+	0.982687i	$0.559318\pi$
$374$		0			0
$375$			1.81156e11i			0.473055i
$376$		0			0
$377$	−2.24098e9	+	4.53506e8i	−0.00571350	+	0.00115624i
$378$		0			0
$379$			4.89603e11i			1.21890i	0.792825	+	0.609449i	$0.208609\pi$
$379$			4.89603e11i			1.21890i	−0.792825	+	0.609449i	$0.791391\pi$
$380$		0			0
$381$	4.29753e11			1.04486
$382$		0			0
$383$		−	2.68583e10i		−	0.0637798i	−0.999491	−	0.0318899i	$-0.989847\pi$
$383$		−	2.68583e10i		−	0.0637798i	0.999491	−	0.0318899i	$-0.0101526\pi$
$384$		0			0
$385$			1.55332e10i			0.0360320i
$386$		0			0
$387$	−5.39567e10			−0.122277
$388$		0			0
$389$	−4.95046e10			−0.109616			−0.0548078	−	0.998497i	$-0.517455\pi$
$389$	−4.95046e10			−0.109616			−0.0548078	+	0.998497i	$0.517455\pi$
$390$		0			0
$391$	−1.26902e11			−0.274583
$392$		0			0
$393$	−1.77735e10			−0.0375843
$394$		0			0
$395$		−	1.77632e11i		−	0.367142i
$396$		0			0
$397$		−	1.10067e11i		−	0.222381i	−0.993799	−	0.111191i	$-0.964534\pi$
$397$		−	1.10067e11i		−	0.222381i	0.993799	−	0.111191i	$-0.0354664\pi$
$398$		0			0
$399$	3.65316e11			0.721590
$400$		0			0
$401$		−	8.15205e11i		−	1.57441i	−0.616693	−	0.787204i	$-0.711528\pi$
$401$		−	8.15205e11i		−	1.57441i	0.616693	−	0.787204i	$-0.288472\pi$
$402$		0			0
$403$	−1.35788e11	−	6.70990e11i	−0.256441	−	1.26719i
$404$		0			0
$405$			1.23427e11i			0.227962i
$406$		0			0
$407$	2.10607e11			0.380451
$408$		0			0
$409$		−	3.35495e11i		−	0.592831i	−0.955059	−	0.296415i	$-0.904209\pi$
$409$		−	3.35495e11i		−	0.592831i	0.955059	−	0.296415i	$-0.0957913\pi$
$410$		0			0
$411$			1.98316e11i			0.342823i
$412$		0			0
$413$	1.56314e11			0.264376
$414$		0			0
$415$	−2.39901e11			−0.397022
$416$		0			0
$417$	1.83575e11			0.297305
$418$		0			0
$419$	−1.15955e12			−1.83792			−0.918962	−	0.394346i	$-0.870971\pi$
$419$	−1.15955e12			−1.83792			−0.918962	+	0.394346i	$0.870971\pi$
$420$		0			0
$421$			1.03177e12i			1.60072i	0.599522	+	0.800359i	$0.295358\pi$
$421$			1.03177e12i			1.60072i	−0.599522	+	0.800359i	$0.704642\pi$
$422$		0			0
$423$			2.65658e9i			0.00403451i
$424$		0			0
$425$	−8.78912e11			−1.30676
$426$		0			0
$427$		−	1.62207e11i		−	0.236126i
$428$		0			0
$429$	3.32898e10	+	1.64500e11i	0.0474518	+	0.234481i
$430$		0			0
$431$		−	1.37520e12i		−	1.91963i	−0.280640	−	0.959813i	$-0.590547\pi$
$431$		−	1.37520e12i		−	1.91963i	0.280640	−	0.959813i	$-0.409453\pi$
$432$		0			0
$433$	−1.22958e12			−1.68097			−0.840487	−	0.541831i	$-0.817731\pi$
$433$	−1.22958e12			−1.68097			−0.840487	+	0.541831i	$0.817731\pi$
$434$		0			0
$435$		−	1.06549e9i		−	0.00142675i
$436$		0			0
$437$		−	2.08312e11i		−	0.273242i
$438$		0			0
$439$	5.10701e11			0.656261			0.328131	−	0.944632i	$-0.393581\pi$
$439$	5.10701e11			0.656261			0.328131	+	0.944632i	$0.393581\pi$
$440$		0			0
$441$	−6.05116e10			−0.0761843
$442$		0			0
$443$	7.52842e11			0.928725			0.464362	−	0.885645i	$-0.346283\pi$
$443$	7.52842e11			0.928725			0.464362	+	0.885645i	$0.346283\pi$
$444$		0			0
$445$	1.14910e11			0.138912
$446$		0			0
$447$			5.67822e11i			0.672711i
$448$		0			0
$449$			1.54365e12i			1.79242i	0.443625	+	0.896212i	$0.353692\pi$
$449$			1.54365e12i			1.79242i	−0.443625	+	0.896212i	$0.646308\pi$
$450$		0			0
$451$	−2.18224e11			−0.248375
$452$		0			0
$453$		−	2.88515e11i		−	0.321904i
$454$		0			0
$455$	−1.27396e11	+	2.57810e10i	−0.139349	+	0.0281999i
$456$		0			0
$457$			4.78491e11i			0.513158i	0.966523	+	0.256579i	$0.0825954\pi$
$457$			4.78491e11i			0.513158i	−0.966523	+	0.256579i	$0.917405\pi$
$458$		0			0
$459$	−1.39456e12			−1.46650
$460$		0			0
$461$			1.49102e12i			1.53755i	0.639517	+	0.768777i	$0.279134\pi$
$461$			1.49102e12i			1.53755i	−0.639517	+	0.768777i	$0.720866\pi$
$462$		0			0
$463$		−	1.01692e12i		−	1.02843i	−0.857662	−	0.514214i	$-0.828084\pi$
$463$		−	1.01692e12i		−	1.02843i	0.857662	−	0.514214i	$-0.171916\pi$
$464$		0			0
$465$	3.19028e11			0.316439
$466$		0			0
$467$	9.10262e11			0.885605			0.442803	−	0.896619i	$-0.353984\pi$
$467$	9.10262e11			0.885605			0.442803	+	0.896619i	$0.353984\pi$
$468$		0			0
$469$	−2.83009e11			−0.270099
$470$		0			0
$471$	1.01336e12			0.948787
$472$		0			0
$473$		−	3.09679e11i		−	0.284470i
$474$		0			0
$475$		−	1.44275e12i		−	1.30038i
$476$		0			0
$477$	−2.19449e10			−0.0194089
$478$		0			0
$479$			2.26651e11i			0.196720i	0.995151	+	0.0983600i	$0.0313597\pi$
$479$			2.26651e11i			0.196720i	−0.995151	+	0.0983600i	$0.968640\pi$
$480$		0			0
$481$	3.49551e11	+	1.72729e12i	0.297754	+	1.47134i
$482$		0			0
$483$			1.21342e11i			0.101449i
$484$		0			0
$485$	3.84431e10			0.0315486
$486$		0			0
$487$		−	2.44539e11i		−	0.197001i	−0.995137	−	0.0985005i	$-0.968595\pi$
$487$		−	2.44539e11i		−	0.197001i	0.995137	−	0.0985005i	$-0.0314046\pi$
$488$		0			0
$489$		−	7.10551e11i		−	0.561960i
$490$		0			0
$491$	−5.96940e11			−0.463515			−0.231758	−	0.972774i	$-0.574448\pi$
$491$	−5.96940e11			−0.463515			−0.231758	+	0.972774i	$0.574448\pi$
$492$		0			0
$493$	1.07115e10			0.00816654
$494$		0			0
$495$	9.56205e9			0.00715859
$496$		0			0
$497$	−1.45403e12			−1.06898
$498$		0			0
$499$		−	1.20878e12i		−	0.872763i	−0.899762	−	0.436382i	$-0.856260\pi$
$499$		−	1.20878e12i		−	0.872763i	0.899762	−	0.436382i	$-0.143740\pi$
$500$		0			0
$501$		−	1.68269e12i		−	1.19325i
$502$		0			0
$503$	−2.40420e12			−1.67461			−0.837306	−	0.546734i	$-0.815871\pi$
$503$	−2.40420e12			−1.67461			−0.837306	+	0.546734i	$0.815871\pi$
$504$		0			0
$505$			7.20114e11i			0.492709i
$506$		0			0
$507$	−1.29389e12	+	5.46051e11i	−0.869686	+	0.367026i
$508$		0			0
$509$		−	9.84554e11i		−	0.650144i	−0.945689	−	0.325072i	$-0.894611\pi$
$509$		−	9.84554e11i		−	0.650144i	0.945689	−	0.325072i	$-0.105389\pi$
$510$		0			0
$511$	1.49280e12			0.968515
$512$		0			0
$513$		−	2.28920e12i		−	1.45934i
$514$		0			0
$515$			4.91733e11i			0.308033i
$516$		0			0
$517$	−1.52471e10			−0.00938600
$518$		0			0
$519$	6.17000e11			0.373278
$520$		0			0
$521$	−1.14661e12			−0.681785			−0.340892	−	0.940102i	$-0.610729\pi$
$521$	−1.14661e12			−0.681785			−0.340892	+	0.940102i	$0.610729\pi$
$522$		0			0
$523$	−2.78844e12			−1.62969			−0.814843	−	0.579682i	$-0.803177\pi$
$523$	−2.78844e12			−1.62969			−0.814843	+	0.579682i	$0.803177\pi$
$524$		0			0
$525$			8.40403e11i			0.482804i
$526$		0			0
$527$			3.20721e12i			1.81125i
$528$		0			0
$529$	−1.73196e12			−0.961585
$530$		0			0
$531$		−	9.62244e10i		−	0.0525243i
$532$		0			0
$533$	−3.62192e11	−	1.78976e12i	−0.194386	−	0.960554i
$534$		0			0
$535$			5.14281e11i			0.271399i
$536$		0			0
$537$	1.26047e12			0.654105
$538$		0			0
$539$		−	3.47300e11i		−	0.177237i
$540$		0			0
$541$			1.86097e12i			0.934013i	0.884254	+	0.467006i	$0.154667\pi$
$541$			1.86097e12i			0.934013i	−0.884254	+	0.467006i	$0.845333\pi$
$542$		0			0
$543$	2.56528e12			1.26629
$544$		0			0
$545$	−6.67936e11			−0.324303
$546$		0			0
$547$	−1.52470e12			−0.728183			−0.364091	−	0.931363i	$-0.618620\pi$
$547$	−1.52470e12			−0.728183			−0.364091	+	0.931363i	$0.618620\pi$
$548$		0			0
$549$	−9.98524e10			−0.0469119
$550$		0			0
$551$			1.75831e10i			0.00812667i
$552$		0			0
$553$		−	1.70750e12i		−	0.776423i
$554$		0			0
$555$	−8.21255e11			−0.367417
$556$		0			0
$557$		−	3.57603e12i		−	1.57418i	−0.616841	−	0.787088i	$-0.711588\pi$
$557$		−	3.57603e12i		−	1.57418i	0.616841	−	0.787088i	$-0.288412\pi$
$558$		0			0
$559$	2.53982e12	−	5.13982e11i	1.10015	−	0.222636i
$560$		0			0
$561$		−	7.86279e11i		−	0.335154i
$562$		0			0
$563$	−3.95357e12			−1.65845			−0.829223	−	0.558918i	$-0.811217\pi$
$563$	−3.95357e12			−1.65845			−0.829223	+	0.558918i	$0.811217\pi$
$564$		0			0
$565$			7.34475e11i			0.303221i
$566$		0			0
$567$			1.18645e12i			0.482088i
$568$		0			0
$569$	2.79161e12			1.11648			0.558238	−	0.829681i	$-0.311478\pi$
$569$	2.79161e12			1.11648			0.558238	+	0.829681i	$0.311478\pi$
$570$		0			0
$571$	−3.50152e12			−1.37846			−0.689230	−	0.724543i	$-0.742051\pi$
$571$	−3.50152e12			−1.37846			−0.689230	+	0.724543i	$0.742051\pi$
$572$		0			0
$573$	3.60579e11			0.139735
$574$		0			0
$575$	4.79219e11			0.182822
$576$		0			0
$577$		−	2.14125e12i		−	0.804223i	−0.915591	−	0.402111i	$-0.868276\pi$
$577$		−	2.14125e12i		−	0.804223i	0.915591	−	0.402111i	$-0.131724\pi$
$578$		0			0
$579$			3.73384e12i			1.38071i
$580$		0			0
$581$	−2.30607e12			−0.839613
$582$		0			0
$583$		−	1.25950e11i		−	0.0451534i
$584$		0			0
$585$	1.58704e10	+	7.84230e10i	0.00560256	+	0.0276848i
$586$		0			0
$587$		−	3.27804e12i		−	1.13958i	−0.821792	−	0.569788i	$-0.807025\pi$
$587$		−	3.27804e12i		−	1.13958i	0.821792	−	0.569788i	$-0.192975\pi$
$588$		0			0
$589$	−5.26469e12			−1.80241
$590$		0			0
$591$			2.49073e12i			0.839815i
$592$		0			0
$593$			3.11914e12i			1.03583i	0.855431	+	0.517916i	$0.173292\pi$
$593$			3.11914e12i			1.03583i	−0.855431	+	0.517916i	$0.826708\pi$
$594$		0			0
$595$	6.08927e11			0.199177
$596$		0			0
$597$	2.23254e12			0.719306
$598$		0			0
$599$	−5.26541e10			−0.0167114			−0.00835568	−	0.999965i	$-0.502660\pi$
$599$	−5.26541e10			−0.0167114			−0.00835568	+	0.999965i	$0.502660\pi$
$600$		0			0
$601$	−9.96737e11			−0.311634			−0.155817	−	0.987786i	$-0.549801\pi$
$601$	−9.96737e11			−0.311634			−0.155817	+	0.987786i	$0.549801\pi$
$602$		0			0
$603$			1.74216e11i			0.0536613i
$604$		0			0
$605$		−	7.99549e11i		−	0.242631i
$606$		0			0
$607$	2.30513e11			0.0689201			0.0344600	−	0.999406i	$-0.489029\pi$
$607$	2.30513e11			0.0689201			0.0344600	+	0.999406i	$0.489029\pi$
$608$		0			0
$609$		−	1.02422e10i		−	0.00301727i
$610$		0			0
$611$	−2.53061e10	−	1.25049e11i	−0.00734580	−	0.0362990i
$612$		0			0
$613$			5.30859e12i			1.51847i	0.650815	+	0.759236i	$0.274427\pi$
$613$			5.30859e12i			1.51847i	−0.650815	+	0.759236i	$0.725573\pi$
$614$		0			0
$615$	8.50954e11			0.239866
$616$		0			0
$617$		−	3.16666e11i		−	0.0879666i	−0.999032	−	0.0439833i	$-0.985995\pi$
$617$		−	3.16666e11i		−	0.0879666i	0.999032	−	0.0439833i	$-0.0140048\pi$
$618$		0			0
$619$		−	1.71107e12i		−	0.468447i	−0.972183	−	0.234223i	$-0.924745\pi$
$619$		−	1.71107e12i		−	0.468447i	0.972183	−	0.234223i	$-0.0752547\pi$
$620$		0			0
$621$	7.60373e11			0.205170
$622$		0			0
$623$	1.10459e12			0.293767
$624$		0			0
$625$	3.06257e12			0.802834
$626$		0			0
$627$	1.29069e12			0.333518
$628$		0			0
$629$		−	8.25614e12i		−	2.10305i
$630$		0			0
$631$		−	1.30123e12i		−	0.326756i	−0.986564	−	0.163378i	$-0.947761\pi$
$631$		−	1.30123e12i		−	0.326756i	0.986564	−	0.163378i	$-0.0522390\pi$
$632$		0			0
$633$	3.23660e12			0.801258
$634$		0			0
$635$			1.17588e12i			0.286999i
$636$		0			0
$637$	2.84837e12	−	5.76423e11i	0.685440	−	0.138712i
$638$		0			0
$639$			8.95081e11i			0.212378i
$640$		0			0
$641$	3.70360e12			0.866488			0.433244	−	0.901277i	$-0.357369\pi$
$641$	3.70360e12			0.866488			0.433244	+	0.901277i	$0.357369\pi$
$642$		0			0
$643$			2.90261e12i			0.669637i	0.942283	+	0.334818i	$0.108675\pi$
$643$			2.90261e12i			0.669637i	−0.942283	+	0.334818i	$0.891325\pi$
$644$		0			0
$645$			1.20758e12i			0.274724i
$646$		0			0
$647$	4.35864e12			0.977871			0.488936	−	0.872320i	$-0.337385\pi$
$647$	4.35864e12			0.977871			0.488936	+	0.872320i	$0.337385\pi$
$648$		0			0
$649$	5.52269e11			0.122194
$650$		0			0
$651$	3.06668e12			0.669198
$652$		0			0
$653$	−4.15712e12			−0.894713			−0.447356	−	0.894356i	$-0.647634\pi$
$653$	−4.15712e12			−0.894713			−0.447356	+	0.894356i	$0.647634\pi$
$654$		0			0
$655$		−	4.86312e10i		−	0.0103236i
$656$		0			0
$657$		−	9.18944e11i		−	0.192418i
$658$		0			0
$659$	−2.27309e12			−0.469496			−0.234748	−	0.972056i	$-0.575426\pi$
$659$	−2.27309e12			−0.469496			−0.234748	+	0.972056i	$0.575426\pi$
$660$		0			0
$661$			8.65901e12i			1.76426i	0.471010	+	0.882128i	$0.343889\pi$
$661$			8.65901e12i			1.76426i	−0.471010	+	0.882128i	$0.656111\pi$
$662$		0			0
$663$	6.44866e12	−	1.30501e12i	1.29616	−	0.262303i
$664$		0			0
$665$			9.99565e11i			0.198205i
$666$		0			0
$667$	−5.84033e9			−0.00114254
$668$		0			0
$669$		−	2.02354e12i		−	0.390566i
$670$		0			0
$671$		−	5.73092e11i		−	0.109137i
$672$		0			0
$673$	6.13539e12			1.15286			0.576428	−	0.817148i	$-0.304446\pi$
$673$	6.13539e12			1.15286			0.576428	+	0.817148i	$0.304446\pi$
$674$		0			0
$675$	5.26627e12			0.976420
$676$		0			0
$677$	8.71612e12			1.59468			0.797341	−	0.603529i	$-0.206239\pi$
$677$	8.71612e12			1.59468			0.797341	+	0.603529i	$0.206239\pi$
$678$		0			0
$679$	3.69538e11			0.0667183
$680$		0			0
$681$			1.31084e12i			0.233554i
$682$		0			0
$683$			3.87456e12i			0.681286i	0.940193	+	0.340643i	$0.110645\pi$
$683$			3.87456e12i			0.681286i	−0.940193	+	0.340643i	$0.889355\pi$
$684$		0			0
$685$	−5.42626e11			−0.0941658
$686$		0			0
$687$			2.14088e12i			0.366679i
$688$		0			0
$689$	1.03298e12	−	2.09043e11i	0.174624	−	0.0353386i
$690$		0			0
$691$		−	1.03381e13i		−	1.72500i	−0.506058	−	0.862499i	$-0.668898\pi$
$691$		−	1.03381e13i		−	1.72500i	0.506058	−	0.862499i	$-0.331102\pi$
$692$		0			0
$693$	9.19161e10			0.0151388
$694$		0			0
$695$			5.02293e11i			0.0816630i
$696$		0			0
$697$			8.55470e12i			1.37296i
$698$		0			0
$699$	−7.37312e12			−1.16816
$700$		0			0
$701$	−3.17236e12			−0.496194			−0.248097	−	0.968735i	$-0.579805\pi$
$701$	−3.17236e12			−0.496194			−0.248097	+	0.968735i	$0.579805\pi$
$702$		0			0
$703$	1.35526e13			2.09278
$704$		0			0
$705$	5.94556e10			0.00906445
$706$		0			0
$707$			6.92217e12i			1.04197i
$708$		0			0
$709$		−	6.29018e12i		−	0.934878i	−0.884025	−	0.467439i	$-0.845177\pi$
$709$		−	6.29018e12i		−	0.934878i	0.884025	−	0.467439i	$-0.154823\pi$
$710$		0			0
$711$	−1.05112e12			−0.154254
$712$		0			0
$713$		−	1.74870e12i		−	0.253403i
$714$		0			0
$715$	−4.50100e11	+	9.10863e10i	−0.0644068	+	0.0130339i
$716$		0			0
$717$			3.26884e12i			0.461910i
$718$		0			0
$719$	9.32507e12			1.30128			0.650642	−	0.759385i	$-0.274500\pi$
$719$	9.32507e12			1.30128			0.650642	+	0.759385i	$0.274500\pi$
$720$		0			0
$721$			4.72683e12i			0.651420i
$722$		0			0
$723$		−	1.25017e13i		−	1.70156i
$724$		0			0
$725$	−4.04496e10			−0.00543742
$726$		0			0
$727$	8.69921e12			1.15498			0.577491	−	0.816397i	$-0.304032\pi$
$727$	8.69921e12			1.15498			0.577491	+	0.816397i	$0.304032\pi$
$728$		0			0
$729$	8.26546e12			1.08391
$730$		0			0
$731$	−1.21399e13			−1.57248
$732$		0			0
$733$			8.52563e12i			1.09083i	0.838165	+	0.545417i	$0.183629\pi$
$733$			8.52563e12i			1.09083i	−0.838165	+	0.545417i	$0.816371\pi$
$734$		0			0
$735$			1.35428e12i			0.171165i
$736$		0			0
$737$	−9.99895e11			−0.124839
$738$		0			0
$739$		−	1.02828e13i		−	1.26827i	−0.773222	−	0.634136i	$-0.781356\pi$
$739$		−	1.02828e13i		−	1.26827i	0.773222	−	0.634136i	$-0.218644\pi$
$740$		0			0
$741$	2.14220e12	+	1.05856e13i	0.261022	+	1.28983i
$742$		0			0
$743$		−	7.34049e12i		−	0.883640i	−0.897104	−	0.441820i	$-0.854333\pi$
$743$		−	7.34049e12i		−	0.883640i	0.897104	−	0.441820i	$-0.145667\pi$
$744$		0			0
$745$	−1.55366e12			−0.184779
$746$		0			0
$747$			1.41958e12i			0.166808i
$748$		0			0
$749$			4.94357e12i			0.573948i
$750$		0			0
$751$	1.32705e13			1.52232			0.761160	−	0.648564i	$-0.224630\pi$
$751$	1.32705e13			1.52232			0.761160	+	0.648564i	$0.224630\pi$
$752$		0			0
$753$	5.73088e12			0.649597
$754$		0			0
$755$	7.89426e11			0.0884199
$756$		0			0
$757$	−3.74741e12			−0.414763			−0.207381	−	0.978260i	$-0.566494\pi$
$757$	−3.74741e12			−0.414763			−0.207381	+	0.978260i	$0.566494\pi$
$758$		0			0
$759$			4.28711e11i			0.0468897i
$760$		0			0
$761$		−	1.27676e13i		−	1.38000i	−0.723811	−	0.689999i	$-0.757611\pi$
$761$		−	1.27676e13i		−	1.38000i	0.723811	−	0.689999i	$-0.242389\pi$
$762$		0			0
$763$	−6.42060e12			−0.685828
$764$		0			0
$765$		−	3.74847e11i		−	0.0395711i
$766$		0			0
$767$	9.16616e11	+	4.52943e12i	0.0956331	+	0.472568i
$768$		0			0
$769$			1.20179e13i			1.23926i	0.784895	+	0.619628i	$0.212717\pi$
$769$			1.20179e13i			1.23926i	−0.784895	+	0.619628i	$0.787283\pi$
$770$		0			0
$771$	−4.67651e12			−0.476626
$772$		0			0
$773$		−	6.82459e12i		−	0.687494i	−0.939062	−	0.343747i	$-0.888304\pi$
$773$		−	6.82459e12i		−	0.687494i	0.939062	−	0.343747i	$-0.111696\pi$
$774$		0			0
$775$		−	1.21113e13i		−	1.20596i
$776$		0			0
$777$	−7.89440e12			−0.777006
$778$		0			0
$779$	−1.40427e13			−1.36626
$780$		0			0
$781$	−5.13722e12			−0.494081
$782$		0			0
$783$	−6.41811e10			−0.00610210
$784$		0			0
$785$			2.77272e12i			0.260611i
$786$		0			0
$787$			1.27629e13i			1.18595i	0.805223	+	0.592973i	$0.202046\pi$
$787$			1.27629e13i			1.18595i	−0.805223	+	0.592973i	$0.797954\pi$
$788$		0			0
$789$	−1.20917e13			−1.11081
$790$		0			0
$791$			7.06021e12i			0.641244i
$792$		0			0
$793$	4.70020e12	−	9.51176e11i	0.422073	−	0.0854145i
$794$		0			0
$795$			4.91138e11i			0.0436065i
$796$		0			0
$797$	5.47156e12			0.480340			0.240170	−	0.970731i	$-0.422797\pi$
$797$	5.47156e12			0.480340			0.240170	+	0.970731i	$0.422797\pi$
$798$		0			0
$799$			5.97711e11i			0.0518837i
$800$		0			0
$801$		−	6.79968e11i		−	0.0583636i
$802$		0			0
$803$	5.27418e12			0.447646
$804$		0			0
$805$	−3.32012e11			−0.0278658
$806$		0			0
$807$	−3.76481e12			−0.312473
$808$		0			0
$809$	−9.52451e12			−0.781762			−0.390881	−	0.920441i	$-0.627830\pi$
$809$	−9.52451e12			−0.781762			−0.390881	+	0.920441i	$0.627830\pi$
$810$		0			0
$811$		−	5.00075e12i		−	0.405921i	−0.979187	−	0.202961i	$-0.934944\pi$
$811$		−	5.00075e12i		−	0.405921i	0.979187	−	0.202961i	$-0.0650563\pi$
$812$		0			0
$813$		−	1.67748e13i		−	1.34664i
$814$		0			0
$815$	1.94419e12			0.154358
$816$		0			0
$817$		−	1.99278e13i		−	1.56481i
$818$		0			0
$819$	1.52556e11	+	7.53848e11i	0.0118482	+	0.0585472i
$820$		0			0
$821$			5.17065e12i			0.397192i	0.980081	+	0.198596i	$0.0636382\pi$
$821$			5.17065e12i			0.397192i	−0.980081	+	0.198596i	$0.936362\pi$
$822$		0			0
$823$	−2.38231e13			−1.81009			−0.905043	−	0.425321i	$-0.860161\pi$
$823$	−2.38231e13			−1.81009			−0.905043	+	0.425321i	$0.860161\pi$
$824$		0			0
$825$			2.96922e12i			0.223151i
$826$		0			0
$827$		−	4.12706e11i		−	0.0306808i	−0.999882	−	0.0153404i	$-0.995117\pi$
$827$		−	4.12706e11i		−	0.0306808i	0.999882	−	0.0153404i	$-0.00488319\pi$
$828$		0			0
$829$	2.06559e13			1.51897			0.759485	−	0.650524i	$-0.225451\pi$
$829$	2.06559e13			1.51897			0.759485	+	0.650524i	$0.225451\pi$
$830$		0			0
$831$	−4.90983e10			−0.00357159
$832$		0			0
$833$	−1.36147e13			−0.979727
$834$		0			0
$835$	4.60411e12			0.327760
$836$		0			0
$837$		−	1.92170e13i		−	1.35338i
$838$		0			0
$839$			2.43146e13i			1.69410i	0.531514	+	0.847049i	$0.321623\pi$
$839$			2.43146e13i			1.69410i	−0.531514	+	0.847049i	$0.678377\pi$
$840$		0			0
$841$	−1.45067e13			−0.999966
$842$		0			0
$843$			1.57425e13i			1.07362i
$844$		0			0
$845$	−1.49409e12	−	3.54031e12i	−0.100814	−	0.238883i
$846$		0			0
$847$		−	7.68575e12i		−	0.513110i
$848$		0			0
$849$	−1.52342e13			−1.00632
$850$		0			0
$851$			4.50158e12i			0.294227i
$852$		0			0
$853$		−	7.91524e12i		−	0.511910i	−0.966689	−	0.255955i	$-0.917610\pi$
$853$		−	7.91524e12i		−	0.511910i	0.966689	−	0.255955i	$-0.0823898\pi$
$854$		0			0
$855$	6.15318e11			0.0393779
$856$		0			0
$857$	2.95462e12			0.187106			0.0935529	−	0.995614i	$-0.470178\pi$
$857$	2.95462e12			0.187106			0.0935529	+	0.995614i	$0.470178\pi$
$858$		0			0
$859$	−1.66946e13			−1.04618			−0.523091	−	0.852277i	$-0.675221\pi$
$859$	−1.66946e13			−1.04618			−0.523091	+	0.852277i	$0.675221\pi$
$860$		0			0
$861$	8.17988e12			0.507263
$862$		0			0
$863$			1.66497e13i			1.02178i	0.859646	+	0.510891i	$0.170684\pi$
$863$			1.66497e13i			1.02178i	−0.859646	+	0.510891i	$0.829316\pi$
$864$		0			0
$865$			1.68822e12i			0.102531i
$866$		0			0
$867$	−1.51183e13			−0.908693
$868$		0			0
$869$		−	6.03276e12i		−	0.358861i
$870$		0			0
$871$	−1.65955e12	−	8.20062e12i	−0.0977034	−	0.482798i
$872$		0			0
$873$		−	2.27482e11i		−	0.0132551i
$874$		0			0
$875$	−4.76470e12			−0.274789
$876$		0			0
$877$		−	1.30129e13i		−	0.742805i	−0.928472	−	0.371402i	$-0.878877\pi$
$877$		−	1.30129e13i		−	0.742805i	0.928472	−	0.371402i	$-0.121123\pi$
$878$		0			0
$879$			2.13925e12i			0.120868i
$880$		0			0
$881$	1.18599e13			0.663268			0.331634	−	0.943408i	$-0.392400\pi$
$881$	1.18599e13			0.663268			0.331634	+	0.943408i	$0.392400\pi$
$882$		0			0
$883$	4.75642e12			0.263304			0.131652	−	0.991296i	$-0.457972\pi$
$883$	4.75642e12			0.263304			0.131652	+	0.991296i	$0.457972\pi$
$884$		0			0
$885$	−2.15355e12			−0.118008
$886$		0			0
$887$	−2.41436e13			−1.30962			−0.654810	−	0.755794i	$-0.727251\pi$
$887$	−2.41436e13			−1.30962			−0.654810	+	0.755794i	$0.727251\pi$
$888$		0			0
$889$			1.13032e13i			0.606939i
$890$		0			0
$891$			4.19184e12i			0.222820i
$892$		0			0
$893$	−9.81153e11			−0.0516304
$894$		0			0
$895$			3.44885e12i			0.179668i
$896$		0			0
$897$	−3.51607e12	+	7.11544e11i	−0.181339	+	0.0366974i
$898$		0			0
$899$			1.47603e11i			0.00753662i
$900$		0			0
$901$	−4.93745e12			−0.249598
$902$		0			0
$903$			1.16080e13i			0.580981i
$904$		0			0
$905$			7.01902e12i			0.347823i
$906$		0			0
$907$	1.16862e13			0.573375			0.286688	−	0.958024i	$-0.407446\pi$
$907$	1.16862e13			0.573375			0.286688	+	0.958024i	$0.407446\pi$
$908$		0			0
$909$	4.26119e12			0.207011
$910$		0			0
$911$	2.77398e13			1.33435			0.667176	−	0.744900i	$-0.267503\pi$
$911$	2.77398e13			1.33435			0.667176	+	0.744900i	$0.267503\pi$
$912$		0			0
$913$	−8.14753e12			−0.388068
$914$		0			0
$915$			2.23475e12i			0.105398i
$916$		0			0
$917$		−	4.67473e11i		−	0.0218320i
$918$		0			0
$919$	1.69347e13			0.783175			0.391588	−	0.920141i	$-0.371926\pi$
$919$	1.69347e13			0.783175			0.391588	+	0.920141i	$0.371926\pi$
$920$		0			0
$921$		−	1.09411e13i		−	0.501063i
$922$		0			0
$923$	−8.52638e12	−	4.21328e13i	−0.386685	−	1.91079i
$924$		0			0
$925$			3.11775e13i			1.40024i
$926$		0			0
$927$	2.90977e12			0.129420
$928$		0			0
$929$		−	3.81216e13i		−	1.67919i	−0.543212	−	0.839595i	$-0.682792\pi$
$929$		−	3.81216e13i		−	1.67919i	0.543212	−	0.839595i	$-0.317208\pi$
$930$		0			0
$931$		−	2.23488e13i		−	0.974944i
$932$		0			0
$933$	−9.84246e12			−0.425242
$934$		0			0
$935$	2.15139e12			0.0920592
$936$		0			0
$937$	1.29409e13			0.548449			0.274225	−	0.961666i	$-0.411579\pi$
$937$	1.29409e13			0.548449			0.274225	+	0.961666i	$0.411579\pi$
$938$		0			0
$939$	−1.81269e13			−0.760903
$940$		0			0
$941$			1.82512e12i			0.0758818i	0.999280	+	0.0379409i	$0.0120799\pi$
$941$			1.82512e12i			0.0758818i	−0.999280	+	0.0379409i	$0.987920\pi$
$942$		0			0
$943$		−	4.66437e12i		−	0.192084i
$944$		0			0
$945$	−3.64858e12			−0.148826
$946$		0			0
$947$			3.80070e13i			1.53564i	0.640668	+	0.767818i	$0.278658\pi$
$947$			3.80070e13i			1.53564i	−0.640668	+	0.767818i	$0.721342\pi$
$948$		0			0
$949$	8.75370e12	+	4.32561e13i	0.350343	+	1.73121i
$950$		0			0
$951$		−	2.75797e13i		−	1.09339i
$952$		0			0
$953$	−3.65722e13			−1.43626			−0.718130	−	0.695909i	$-0.755001\pi$
$953$	−3.65722e13			−1.43626			−0.718130	+	0.695909i	$0.755001\pi$
$954$		0			0
$955$			9.86605e11i			0.0383820i
$956$		0			0
$957$		−	3.61864e10i		−	0.00139458i
$958$		0			0
$959$	−5.21605e12			−0.199140
$960$		0			0
$961$	−1.77554e13			−0.671544
$962$		0			0
$963$	3.04319e12			0.114028
$964$		0			0
$965$	−1.02164e13			−0.379249
$966$		0			0
$967$		−	3.14472e13i		−	1.15655i	−0.815843	−	0.578273i	$-0.803727\pi$
$967$		−	3.14472e13i		−	1.15655i	0.815843	−	0.578273i	$-0.196273\pi$
$968$		0			0
$969$		−	5.05971e13i		−	1.84361i
$970$		0			0
$971$	−3.45091e13			−1.24580			−0.622898	−	0.782303i	$-0.714045\pi$
$971$	−3.45091e13			−1.24580			−0.622898	+	0.782303i	$0.714045\pi$
$972$		0			0
$973$			4.82834e12i			0.172699i
$974$		0			0
$975$	−2.43520e13	+	4.92809e12i	−0.863005	+	0.174646i
$976$		0			0
$977$			2.68549e13i			0.942970i	0.881874	+	0.471485i	$0.156282\pi$
$977$			2.68549e13i			0.942970i	−0.881874	+	0.471485i	$0.843718\pi$
$978$		0			0
$979$	3.90260e12			0.135779
$980$		0			0
$981$			3.95243e12i			0.136255i
$982$		0			0
$983$		−	1.01518e13i		−	0.346780i	−0.984853	−	0.173390i	$-0.944528\pi$
$983$		−	1.01518e13i		−	0.346780i	0.984853	−	0.173390i	$-0.0554721\pi$
$984$		0			0
$985$	−6.81507e12			−0.230678
$986$		0			0
$987$	5.71523e11			0.0191693
$988$		0			0
$989$	6.61915e12			0.219998
$990$		0			0
$991$	1.99349e13			0.656573			0.328287	−	0.944578i	$-0.393529\pi$
$991$	1.99349e13			0.656573			0.328287	+	0.944578i	$0.393529\pi$
$992$		0			0
$993$			2.43242e13i			0.793903i
$994$		0			0
$995$			6.10859e12i			0.197577i
$996$		0			0
$997$	3.87311e13			1.24146			0.620728	−	0.784026i	$-0.286837\pi$
$997$	3.87311e13			1.24146			0.620728	+	0.784026i	$0.286837\pi$
$998$		0			0
$999$			4.94692e13i			1.57141i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.f.d.129.9		32
4.3	odd	2		104.10.f.a.25.24	yes	32
13.12	even	2	inner	208.10.f.d.129.10		32
52.51	odd	2		104.10.f.a.25.23	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.23	✓	32	52.51	odd	2
104.10.f.a.25.24	yes	32	4.3	odd	2
208.10.f.d.129.9		32	1.1	even	1	trivial
208.10.f.d.129.10		32	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-132.434 q^{3} -362.362i q^{5} -3483.24i q^{7} -2144.23 q^{9} -12306.6i q^{11} +(100932. - 20425.6i) q^{13} +47989.0i q^{15} -482437. q^{17} -791929. i q^{19} +461299. i q^{21} +263044. q^{23} +1.82182e6 q^{25} +2.89067e6 q^{27} -22202.9 q^{29} -6.64793e6i q^{31} +1.62981e6i q^{33} -1.26219e6 q^{35} +1.71134e7i q^{37} +(-1.33668e7 + 2.70504e6i) q^{39} -1.77323e7i q^{41} +2.51637e7 q^{43} +776987. i q^{45} -1.23894e6i q^{47} +2.82207e7 q^{49} +6.38910e7 q^{51} +1.02344e7 q^{53} -4.45943e6 q^{55} +1.04878e8i q^{57} +4.48760e7i q^{59} +4.65679e7 q^{61} +7.46886e6i q^{63} +(-7.40144e6 - 3.65739e7i) q^{65} -8.12489e7i q^{67} -3.48360e7 q^{69} -4.17437e8i q^{71} +4.28566e8i q^{73} -2.41271e8 q^{75} -4.28667e7 q^{77} +4.90206e8 q^{79} -3.40618e8 q^{81} -6.62047e8i q^{83} +1.74817e8i q^{85} +2.94041e6 q^{87} +3.17115e8i q^{89} +(-7.11470e7 - 3.51570e8i) q^{91} +8.80412e8i q^{93} -2.86964e8 q^{95} +1.06090e8i q^{97} +2.63881e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) \) 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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