LMFDB - Embedded newform 912.2.d.a.191.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,2,Mod(191,912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("912.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.2593100598870016.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.12
Root			$1.37027 - 0.349801i$ of defining polynomial
Character	$\chi$	$=$	912.191
Dual form			912.2.d.a.191.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.72007 + 0.203364i) q^{3} +2.74054i q^{5} +1.91729i q^{7} +(2.91729 + 0.699602i) q^{9} -0.463141 q^{11} +0.406728 q^{13} +(-0.557328 + 4.71392i) q^{15} -0.415054i q^{17} -1.00000i q^{19} +(-0.389907 + 3.29787i) q^{21} -2.91913 q^{23} -2.51056 q^{25} +(4.87566 + 1.79664i) q^{27} +7.11674i q^{29} -2.40673i q^{31} +(-0.796636 - 0.0941864i) q^{33} -5.25440 q^{35} -6.61439 q^{37} +(0.699602 + 0.0827140i) q^{39} +5.43299i q^{41} -5.91729i q^{43} +(-1.91729 + 7.99494i) q^{45} +3.61873 q^{47} +3.32401 q^{49} +(0.0844071 - 0.713922i) q^{51} +9.15768i q^{53} -1.26926i q^{55} +(0.203364 - 1.72007i) q^{57} -7.99494 q^{59} +8.15859 q^{61} +(-1.34134 + 5.59327i) q^{63} +1.11466i q^{65} -14.8557i q^{67} +(-5.02112 - 0.593647i) q^{69} +9.75133 q^{71} -1.13747 q^{73} +(-4.31834 - 0.510558i) q^{75} -0.887974i q^{77} -13.8346i q^{79} +(8.02112 + 4.08188i) q^{81} +6.54765 q^{83} +1.13747 q^{85} +(-1.44729 + 12.2413i) q^{87} -6.83219i q^{89} +0.779815i q^{91} +(0.489442 - 4.13974i) q^{93} +2.74054 q^{95} +11.2624 q^{97} +(-1.35112 - 0.324014i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.72007	+	0.203364i	0.993083	+	0.117412i
$3$	1.72007	+	0.203364i	0.993083	+	0.117412i
$4$		0			0
$5$			2.74054i			1.22561i	0.790235	+	0.612803i	$0.209958\pi$
$5$			2.74054i			1.22561i	−0.790235	+	0.612803i	$0.790042\pi$
$6$		0			0
$7$			1.91729i			0.724666i	0.932049	+	0.362333i	$0.118020\pi$
$7$			1.91729i			0.724666i	−0.932049	+	0.362333i	$0.881980\pi$
$8$		0			0
$9$	2.91729	+	0.699602i	0.972429	+	0.233201i
$10$		0			0
$11$	−0.463141			−0.139642			−0.0698212	−	0.997560i	$-0.522243\pi$
$11$	−0.463141			−0.139642			−0.0698212	+	0.997560i	$0.522243\pi$
$12$		0			0
$13$	0.406728			0.112806			0.0564031	−	0.998408i	$-0.482037\pi$
$13$	0.406728			0.112806			0.0564031	+	0.998408i	$0.482037\pi$
$14$		0			0
$15$	−0.557328	+	4.71392i	−0.143901	+	1.21713i
$16$		0			0
$17$		−	0.415054i		−	0.100665i	−0.998733	−	0.0503327i	$-0.983972\pi$
$17$		−	0.415054i		−	0.100665i	0.998733	−	0.0503327i	$-0.0160282\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$	−0.389907	+	3.29787i	−0.0850848	+	0.719654i
$22$		0			0
$23$	−2.91913			−0.608681			−0.304341	−	0.952563i	$-0.598436\pi$
$23$	−2.91913			−0.608681			−0.304341	+	0.952563i	$0.598436\pi$
$24$		0			0
$25$	−2.51056			−0.502112
$26$		0			0
$27$	4.87566	+	1.79664i	0.938322	+	0.345763i
$28$		0			0
$29$			7.11674i			1.32155i	0.750586	+	0.660773i	$0.229771\pi$
$29$			7.11674i			1.32155i	−0.750586	+	0.660773i	$0.770229\pi$
$30$		0			0
$31$		−	2.40673i		−	0.432261i	−0.976364	−	0.216131i	$-0.930656\pi$
$31$		−	2.40673i		−	0.432261i	0.976364	−	0.216131i	$-0.0693437\pi$
$32$		0			0
$33$	−0.796636	−	0.0941864i	−0.138676	−	0.0163957i
$34$		0			0
$35$	−5.25440			−0.888155
$36$		0			0
$37$	−6.61439			−1.08740			−0.543699	−	0.839280i	$-0.682977\pi$
$37$	−6.61439			−1.08740			−0.543699	+	0.839280i	$0.682977\pi$
$38$		0			0
$39$	0.699602	+	0.0827140i	0.112026	+	0.0132448i
$40$		0			0
$41$			5.43299i			0.848491i	0.905547	+	0.424245i	$0.139461\pi$
$41$			5.43299i			0.848491i	−0.905547	+	0.424245i	$0.860539\pi$
$42$		0			0
$43$		−	5.91729i		−	0.902378i	−0.892429	−	0.451189i	$-0.851000\pi$
$43$		−	5.91729i		−	0.902378i	0.892429	−	0.451189i	$-0.149000\pi$
$44$		0			0
$45$	−1.91729	+	7.99494i	−0.285812	+	1.19181i
$46$		0			0
$47$	3.61873			0.527847			0.263923	−	0.964544i	$-0.414983\pi$
$47$	3.61873			0.527847			0.263923	+	0.964544i	$0.414983\pi$
$48$		0			0
$49$	3.32401			0.474859
$50$		0			0
$51$	0.0844071	−	0.713922i	0.0118194	−	0.0999690i
$52$		0			0
$53$			9.15768i			1.25790i	0.777444	+	0.628952i	$0.216516\pi$
$53$			9.15768i			1.25790i	−0.777444	+	0.628952i	$0.783484\pi$
$54$		0			0
$55$		−	1.26926i		−	0.171147i
$56$		0			0
$57$	0.203364	−	1.72007i	0.0269362	−	0.227829i
$58$		0			0
$59$	−7.99494			−1.04085			−0.520426	−	0.853907i	$-0.674227\pi$
$59$	−7.99494			−1.04085			−0.520426	+	0.853907i	$0.674227\pi$
$60$		0			0
$61$	8.15859			1.04460			0.522300	−	0.852762i	$-0.325074\pi$
$61$	8.15859			1.04460			0.522300	+	0.852762i	$0.325074\pi$
$62$		0			0
$63$	−1.34134	+	5.59327i	−0.168993	+	0.704686i
$64$		0			0
$65$			1.11466i			0.138256i
$66$		0			0
$67$		−	14.8557i		−	1.81491i	−0.420148	−	0.907456i	$-0.638022\pi$
$67$		−	14.8557i		−	1.81491i	0.420148	−	0.907456i	$-0.361978\pi$
$68$		0			0
$69$	−5.02112	−	0.593647i	−0.604471	−	0.0714667i
$70$		0			0
$71$	9.75133			1.15727			0.578635	−	0.815587i	$-0.303586\pi$
$71$	9.75133			1.15727			0.578635	+	0.815587i	$0.303586\pi$
$72$		0			0
$73$	−1.13747			−0.133131			−0.0665655	−	0.997782i	$-0.521204\pi$
$73$	−1.13747			−0.133131			−0.0665655	+	0.997782i	$0.521204\pi$
$74$		0			0
$75$	−4.31834	−	0.510558i	−0.498639	−	0.0589541i
$76$		0			0
$77$		−	0.887974i		−	0.101194i
$78$		0			0
$79$		−	13.8346i		−	1.55651i	−0.627948	−	0.778255i	$-0.716105\pi$
$79$		−	13.8346i		−	1.55651i	0.627948	−	0.778255i	$-0.283895\pi$
$80$		0			0
$81$	8.02112	+	4.08188i	0.891235	+	0.453542i
$82$		0			0
$83$	6.54765			0.718698			0.359349	−	0.933203i	$-0.382999\pi$
$83$	6.54765			0.718698			0.359349	+	0.933203i	$0.382999\pi$
$84$		0			0
$85$	1.13747			0.123376
$86$		0			0
$87$	−1.44729	+	12.2413i	−0.155166	+	1.31241i
$88$		0			0
$89$		−	6.83219i		−	0.724211i	−0.932137	−	0.362106i	$-0.882058\pi$
$89$		−	6.83219i		−	0.724211i	0.932137	−	0.362106i	$-0.117942\pi$
$90$		0			0
$91$			0.779815i			0.0817468i
$92$		0			0
$93$	0.489442	−	4.13974i	0.0507528	−	0.429271i
$94$		0			0
$95$	2.74054			0.281173
$96$		0			0
$97$	11.2624			1.14353			0.571763	−	0.820419i	$-0.306260\pi$
$97$	11.2624			1.14353			0.571763	+	0.820419i	$0.306260\pi$
$98$		0			0
$99$	−1.35112	−	0.324014i	−0.135792	−	0.0325647i
$100$		0			0
$101$		−	1.75639i		−	0.174767i	−0.996175	−	0.0873837i	$-0.972149\pi$
$101$		−	1.75639i		−	0.174767i	0.996175	−	0.0873837i	$-0.0278506\pi$
$102$		0			0
$103$			8.85569i			0.872577i	0.899807	+	0.436288i	$0.143707\pi$
$103$			8.85569i			0.872577i	−0.899807	+	0.436288i	$0.856293\pi$
$104$		0			0
$105$	−9.03794	−	1.06856i	−0.882012	−	0.104280i
$106$		0			0
$107$	6.05018			0.584892			0.292446	−	0.956282i	$-0.405531\pi$
$107$	6.05018			0.584892			0.292446	+	0.956282i	$0.405531\pi$
$108$		0			0
$109$	9.83457			0.941981			0.470991	−	0.882138i	$-0.343897\pi$
$109$	9.83457			0.941981			0.470991	+	0.882138i	$0.343897\pi$
$110$		0			0
$111$	−11.3772	−	1.34513i	−1.07988	−	0.127674i
$112$		0			0
$113$		−	12.4821i		−	1.17422i	−0.809509	−	0.587108i	$-0.800266\pi$
$113$		−	12.4821i		−	1.17422i	0.809509	−	0.587108i	$-0.199734\pi$
$114$		0			0
$115$		−	8.00000i		−	0.746004i
$116$		0			0
$117$	1.18654	+	0.284548i	0.109696	+	0.0263065i
$118$		0			0
$119$	0.795777			0.0729487
$120$		0			0
$121$	−10.7855			−0.980500
$122$		0			0
$123$	−1.10488	+	9.34513i	−0.0996233	+	0.842622i
$124$		0			0
$125$			6.82242i			0.610215i
$126$		0			0
$127$		−	15.0211i		−	1.33291i	−0.745546	−	0.666454i	$-0.767811\pi$
$127$		−	15.0211i		−	1.33291i	0.745546	−	0.666454i	$-0.232189\pi$
$128$		0			0
$129$	1.20336	−	10.1781i	0.105950	−	0.896136i
$130$		0			0
$131$	−14.2237			−1.24273			−0.621365	−	0.783521i	$-0.713422\pi$
$131$	−14.2237			−1.24273			−0.621365	+	0.783521i	$0.713422\pi$
$132$		0			0
$133$	1.91729			0.166250
$134$		0			0
$135$	−4.92375	+	13.3620i	−0.423769	+	1.15001i
$136$		0			0
$137$		−	9.26364i		−	0.791446i	−0.918370	−	0.395723i	$-0.870494\pi$
$137$		−	9.26364i		−	0.791446i	0.918370	−	0.395723i	$-0.129506\pi$
$138$		0			0
$139$			19.1461i			1.62395i	0.583694	+	0.811974i	$0.301607\pi$
$139$			19.1461i			1.62395i	−0.583694	+	0.811974i	$0.698393\pi$
$140$		0			0
$141$	6.22448	+	0.735921i	0.524196	+	0.0619757i
$142$		0			0
$143$	−0.188373			−0.0157525
$144$		0			0
$145$	−19.5037			−1.61970
$146$		0			0
$147$	5.71754	+	0.675986i	0.471575	+	0.0557544i
$148$		0			0
$149$			13.6065i			1.11469i	0.830281	+	0.557345i	$0.188180\pi$
$149$			13.6065i			1.11469i	−0.830281	+	0.557345i	$0.811820\pi$
$150$		0			0
$151$		−	10.7798i		−	0.877249i	−0.898670	−	0.438624i	$-0.855466\pi$
$151$		−	10.7798i		−	0.877249i	0.898670	−	0.438624i	$-0.144534\pi$
$152$		0			0
$153$	0.290372	−	1.21083i	0.0234752	−	0.0978898i
$154$		0			0
$155$	6.59573			0.529782
$156$		0			0
$157$	10.0422			0.801457			0.400729	−	0.916197i	$-0.368757\pi$
$157$	10.0422			0.801457			0.400729	+	0.916197i	$0.368757\pi$
$158$		0			0
$159$	−1.86234	+	15.7519i	−0.147694	+	1.24920i
$160$		0			0
$161$		−	5.59681i		−	0.441091i
$162$		0			0
$163$			16.4826i			1.29102i	0.763753	+	0.645508i	$0.223354\pi$
$163$			16.4826i			1.29102i	−0.763753	+	0.645508i	$0.776646\pi$
$164$		0			0
$165$	0.258121	−	2.18321i	0.0200947	−	0.169963i
$166$		0			0
$167$	6.42692			0.497330			0.248665	−	0.968590i	$-0.420008\pi$
$167$	6.42692			0.497330			0.248665	+	0.968590i	$0.420008\pi$
$168$		0			0
$169$	−12.8346			−0.987275
$170$		0			0
$171$	0.699602	−	2.91729i	0.0534999	−	0.223090i
$172$		0			0
$173$		−	22.6337i		−	1.72081i	−0.509612	−	0.860404i	$-0.670211\pi$
$173$		−	22.6337i		−	1.72081i	0.509612	−	0.860404i	$-0.329789\pi$
$174$		0			0
$175$		−	4.81346i		−	0.363863i
$176$		0			0
$177$	−13.7519	−	1.62588i	−1.03365	−	0.122209i
$178$		0			0
$179$	17.3891			1.29972			0.649860	−	0.760054i	$-0.274827\pi$
$179$	17.3891			1.29972			0.649860	+	0.760054i	$0.274827\pi$
$180$		0			0
$181$	−17.3383			−1.28874			−0.644372	−	0.764712i	$-0.722881\pi$
$181$	−17.3383			−1.28874			−0.644372	+	0.764712i	$0.722881\pi$
$182$		0			0
$183$	14.0333	+	1.65916i	1.03737	+	0.122649i
$184$		0			0
$185$		−	18.1270i		−	1.33272i
$186$		0			0
$187$			0.192229i			0.0140571i
$188$		0			0
$189$	−3.44466	+	9.34804i	−0.250562	+	0.679970i
$190$		0			0
$191$	5.94422			0.430109			0.215054	−	0.976602i	$-0.431007\pi$
$191$	5.94422			0.430109			0.215054	+	0.976602i	$0.431007\pi$
$192$		0			0
$193$	4.20766			0.302874			0.151437	−	0.988467i	$-0.451610\pi$
$193$	4.20766			0.302874			0.151437	+	0.988467i	$0.451610\pi$
$194$		0			0
$195$	−0.226681	+	1.91729i	−0.0162330	+	0.137300i
$196$		0			0
$197$		−	1.28347i		−	0.0914434i	−0.998954	−	0.0457217i	$-0.985441\pi$
$197$		−	1.28347i		−	0.0914434i	0.998954	−	0.0457217i	$-0.0145587\pi$
$198$		0			0
$199$		−	9.75186i		−	0.691291i	−0.938365	−	0.345645i	$-0.887660\pi$
$199$		−	9.75186i		−	0.691291i	0.938365	−	0.345645i	$-0.112340\pi$
$200$		0			0
$201$	3.02112	−	25.5528i	0.213093	−	1.80236i
$202$		0			0
$203$	−13.6448			−0.957679
$204$		0			0
$205$	−14.8893			−1.03992
$206$		0			0
$207$	−8.51595	−	2.04223i	−0.591899	−	0.141945i
$208$		0			0
$209$			0.463141i			0.0320362i
$210$		0			0
$211$		−	11.4701i		−	0.789632i	−0.918760	−	0.394816i	$-0.870808\pi$
$211$		−	11.4701i		−	0.789632i	0.918760	−	0.394816i	$-0.129192\pi$
$212$		0			0
$213$	16.7730	+	1.98307i	1.14927	+	0.135878i
$214$		0			0
$215$	16.2166			1.10596
$216$		0			0
$217$	4.61439			0.313245
$218$		0			0
$219$	−1.95653	−	0.231321i	−0.132210	−	0.0156312i
$220$		0			0
$221$		−	0.168814i		−	0.0113557i
$222$		0			0
$223$			2.44896i			0.163994i	0.996633	+	0.0819972i	$0.0261299\pi$
$223$			2.44896i			0.163994i	−0.996633	+	0.0819972i	$0.973870\pi$
$224$		0			0
$225$	−7.32401	−	1.75639i	−0.488268	−	0.117093i
$226$		0			0
$227$	−20.8292			−1.38248			−0.691242	−	0.722623i	$-0.742936\pi$
$227$	−20.8292			−1.38248			−0.691242	+	0.722623i	$0.742936\pi$
$228$		0			0
$229$	10.9720			0.725053			0.362527	−	0.931973i	$-0.381914\pi$
$229$	10.9720			0.725053			0.362527	+	0.931973i	$0.381914\pi$
$230$		0			0
$231$	0.180582	−	1.52738i	0.0118814	−	0.100494i
$232$		0			0
$233$		−	24.7806i		−	1.62343i	−0.584053	−	0.811715i	$-0.698534\pi$
$233$		−	24.7806i		−	1.62343i	0.584053	−	0.811715i	$-0.301466\pi$
$234$		0			0
$235$			9.91729i			0.646932i
$236$		0			0
$237$	2.81346	−	23.7964i	0.182754	−	1.54574i
$238$		0			0
$239$	−25.3016			−1.63662			−0.818312	−	0.574774i	$-0.805090\pi$
$239$	−25.3016			−1.63662			−0.818312	+	0.574774i	$0.805090\pi$
$240$		0			0
$241$	13.8009			0.888996			0.444498	−	0.895780i	$-0.353382\pi$
$241$	13.8009			0.888996			0.444498	+	0.895780i	$0.353382\pi$
$242$		0			0
$243$	12.9668	+	8.65232i	0.831819	+	0.555047i
$244$		0			0
$245$			9.10959i			0.581991i
$246$		0			0
$247$		−	0.406728i		−	0.0258795i
$248$		0			0
$249$	11.2624	+	1.33156i	0.713727	+	0.0843840i
$250$		0			0
$251$	9.09981			0.574375			0.287188	−	0.957874i	$-0.407280\pi$
$251$	9.09981			0.574375			0.287188	+	0.957874i	$0.407280\pi$
$252$		0			0
$253$	1.35197			0.0849977
$254$		0			0
$255$	1.95653	+	0.231321i	0.122523	+	0.0144859i
$256$		0			0
$257$		−	9.70324i		−	0.605272i	−0.953106	−	0.302636i	$-0.902133\pi$
$257$		−	9.70324i		−	0.605272i	0.953106	−	0.302636i	$-0.0978666\pi$
$258$		0			0
$259$		−	12.6817i		−	0.788001i
$260$		0			0
$261$	−4.97888	+	20.7616i	−0.308185	+	1.28511i
$262$		0			0
$263$	−18.7785			−1.15793			−0.578966	−	0.815352i	$-0.696544\pi$
$263$	−18.7785			−1.15793			−0.578966	+	0.815352i	$0.696544\pi$
$264$		0			0
$265$	−25.0970			−1.54170
$266$		0			0
$267$	1.38942	−	11.7519i	0.0850314	−	0.719202i
$268$		0			0
$269$		−	5.07580i		−	0.309477i	−0.987955	−	0.154739i	$-0.950546\pi$
$269$		−	5.07580i		−	0.309477i	0.987955	−	0.154739i	$-0.0494535\pi$
$270$		0			0
$271$		−	28.1517i		−	1.71010i	−0.518548	−	0.855048i	$-0.673527\pi$
$271$		−	28.1517i		−	1.71010i	0.518548	−	0.855048i	$-0.326473\pi$
$272$		0			0
$273$	−0.158586	+	1.34134i	−0.00959809	+	0.0811814i
$274$		0			0
$275$	1.16274			0.0701160
$276$		0			0
$277$	10.1586			0.610370			0.305185	−	0.952293i	$-0.401282\pi$
$277$	10.1586			0.610370			0.305185	+	0.952293i	$0.401282\pi$
$278$		0			0
$279$	1.68375	−	7.02112i	0.100804	−	0.420343i
$280$		0			0
$281$		−	9.34605i		−	0.557539i	−0.960358	−	0.278769i	$-0.910073\pi$
$281$		−	9.34605i		−	0.557539i	0.960358	−	0.278769i	$-0.0899265\pi$
$282$		0			0
$283$			13.1711i			0.782941i	0.920191	+	0.391471i	$0.128034\pi$
$283$			13.1711i			0.782941i	−0.920191	+	0.391471i	$0.871966\pi$
$284$		0			0
$285$	4.71392	+	0.557328i	0.279229	+	0.0330132i
$286$		0			0
$287$	−10.4166			−0.614872
$288$		0			0
$289$	16.8277			0.989866
$290$		0			0
$291$	19.3722	+	2.29037i	1.13562	+	0.134264i
$292$		0			0
$293$		−	3.86497i		−	0.225794i	−0.993607	−	0.112897i	$-0.963987\pi$
$293$		−	3.86497i		−	0.225794i	0.993607	−	0.112897i	$-0.0360130\pi$
$294$		0			0
$295$		−	21.9104i		−	1.27568i
$296$		0			0
$297$	−2.25812	−	0.832096i	−0.131029	−	0.0482831i
$298$		0			0
$299$	−1.18729			−0.0686630
$300$		0			0
$301$	11.3451			0.653922
$302$		0			0
$303$	0.357187	−	3.02112i	0.0205199	−	0.173559i
$304$		0			0
$305$			22.3589i			1.28027i
$306$		0			0
$307$			2.16543i			0.123588i	0.998089	+	0.0617938i	$0.0196821\pi$
$307$			2.16543i			0.123588i	−0.998089	+	0.0617938i	$0.980318\pi$
$308$		0			0
$309$	−1.80093	+	15.2324i	−0.102451	+	0.866541i
$310$		0			0
$311$	−30.5708			−1.73351			−0.866755	−	0.498735i	$-0.833798\pi$
$311$	−30.5708			−1.73351			−0.866755	+	0.498735i	$0.833798\pi$
$312$		0			0
$313$	24.0845			1.36133			0.680667	−	0.732593i	$-0.261690\pi$
$313$	24.0845			1.36133			0.680667	+	0.732593i	$0.261690\pi$
$314$		0			0
$315$	−15.3286	−	3.67599i	−0.863668	−	0.207118i
$316$		0			0
$317$			9.98779i			0.560970i	0.959858	+	0.280485i	$0.0904953\pi$
$317$			9.98779i			0.560970i	−0.959858	+	0.280485i	$0.909505\pi$
$318$		0			0
$319$		−	3.29606i		−	0.184544i
$320$		0			0
$321$	10.4067	+	1.23039i	0.580847	+	0.0686736i
$322$		0			0
$323$	−0.415054			−0.0230942
$324$		0			0
$325$	−1.02112			−0.0566413
$326$		0			0
$327$	16.9162	+	2.00000i	0.935466	+	0.110600i
$328$		0			0
$329$			6.93815i			0.382513i
$330$		0			0
$331$			6.58074i			0.361710i	0.983510	+	0.180855i	$0.0578865\pi$
$331$			6.58074i			0.361710i	−0.983510	+	0.180855i	$0.942113\pi$
$332$		0			0
$333$	−19.2961	−	4.62744i	−1.05742	−	0.253582i
$334$		0			0
$335$	40.7126			2.22437
$336$		0			0
$337$	5.22019			0.284362			0.142181	−	0.989841i	$-0.454589\pi$
$337$	5.22019			0.284362			0.142181	+	0.989841i	$0.454589\pi$
$338$		0			0
$339$	2.53841	−	21.4701i	0.137868	−	1.16609i
$340$		0			0
$341$			1.11466i			0.0603620i
$342$		0			0
$343$			19.7941i			1.06878i
$344$		0			0
$345$	1.62691	−	13.7606i	0.0875901	−	0.740844i
$346$		0			0
$347$	19.4929			1.04643			0.523216	−	0.852200i	$-0.324732\pi$
$347$	19.4929			1.04643			0.523216	+	0.852200i	$0.324732\pi$
$348$		0			0
$349$	−29.7604			−1.59304			−0.796520	−	0.604612i	$-0.793328\pi$
$349$	−29.7604			−1.59304			−0.796520	+	0.604612i	$0.793328\pi$
$350$		0			0
$351$	1.98307	+	0.730743i	0.105849	+	0.0390042i
$352$		0			0
$353$			19.0297i			1.01285i	0.862284	+	0.506425i	$0.169033\pi$
$353$			19.0297i			1.01285i	−0.862284	+	0.506425i	$0.830967\pi$
$354$		0			0
$355$			26.7239i			1.41836i
$356$		0			0
$357$	1.36879	+	0.161833i	0.0724442	+	0.00856508i
$358$		0			0
$359$	−33.4654			−1.76623			−0.883117	−	0.469153i	$-0.844559\pi$
$359$	−33.4654			−1.76623			−0.883117	+	0.469153i	$0.844559\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	−18.5518	−	2.19338i	−0.973718	−	0.115123i
$364$		0			0
$365$		−	3.11729i		−	0.163166i
$366$		0			0
$367$		−	20.0000i		−	1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$		−	20.0000i		−	1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$		0			0
$369$	−3.80093	+	15.8496i	−0.197869	+	0.825097i
$370$		0			0
$371$	−17.5579			−0.911560
$372$		0			0
$373$	−24.6903			−1.27841			−0.639207	−	0.769035i	$-0.720737\pi$
$373$	−24.6903			−1.27841			−0.639207	+	0.769035i	$0.720737\pi$
$374$		0			0
$375$	−1.38744	+	11.7350i	−0.0716468	+	0.605995i
$376$		0			0
$377$			2.89458i			0.149079i
$378$		0			0
$379$			20.4404i			1.04995i	0.851117	+	0.524976i	$0.175926\pi$
$379$			20.4404i			1.04995i	−0.851117	+	0.524976i	$0.824074\pi$
$380$		0			0
$381$	3.05476	−	25.8374i	0.156500	−	1.32369i
$382$		0			0
$383$	−2.75531			−0.140790			−0.0703949	−	0.997519i	$-0.522426\pi$
$383$	−2.75531			−0.140790			−0.0703949	+	0.997519i	$0.522426\pi$
$384$		0			0
$385$	2.43353			0.124024
$386$		0			0
$387$	4.13974	−	17.2624i	0.210435	−	0.877498i
$388$		0			0
$389$			29.4316i			1.49224i	0.665812	+	0.746120i	$0.268085\pi$
$389$			29.4316i			1.49224i	−0.665812	+	0.746120i	$0.731915\pi$
$390$		0			0
$391$			1.21160i			0.0612731i
$392$		0			0
$393$	−24.4658	−	2.89259i	−1.23414	−	0.145912i
$394$		0			0
$395$	37.9142			1.90767
$396$		0			0
$397$	−11.3029			−0.567276			−0.283638	−	0.958931i	$-0.591541\pi$
$397$	−11.3029			−0.567276			−0.283638	+	0.958931i	$0.591541\pi$
$398$		0			0
$399$	3.29787	+	0.389907i	0.165100	+	0.0195198i
$400$		0			0
$401$			2.35004i			0.117355i	0.998277	+	0.0586776i	$0.0186884\pi$
$401$			2.35004i			0.117355i	−0.998277	+	0.0586776i	$0.981312\pi$
$402$		0			0
$403$		−	0.978885i		−	0.0487617i
$404$		0			0
$405$	−11.1865	+	21.9822i	−0.555864	+	1.09230i
$406$		0			0
$407$	3.06340			0.151847
$408$		0			0
$409$	13.2202			0.653696			0.326848	−	0.945077i	$-0.394013\pi$
$409$	13.2202			0.653696			0.326848	+	0.945077i	$0.394013\pi$
$410$		0			0
$411$	1.88389	−	15.9341i	0.0929255	−	0.785972i
$412$		0			0
$413$		−	15.3286i		−	0.754270i
$414$		0			0
$415$			17.9441i			0.880841i
$416$		0			0
$417$	−3.89362	+	32.9326i	−0.190672	+	1.61272i
$418$		0			0
$419$	4.43407			0.216618			0.108309	−	0.994117i	$-0.465456\pi$
$419$	4.43407			0.216618			0.108309	+	0.994117i	$0.465456\pi$
$420$		0			0
$421$	−12.0086			−0.585263			−0.292631	−	0.956225i	$-0.594531\pi$
$421$	−12.0086			−0.585263			−0.292631	+	0.956225i	$0.594531\pi$
$422$		0			0
$423$	10.5569	+	2.53167i	0.513293	+	0.123094i
$424$		0			0
$425$			1.04202i			0.0505452i
$426$		0			0
$427$			15.6423i			0.756986i
$428$		0			0
$429$	−0.324014	−	0.0383083i	−0.0156436	−	0.00184954i
$430$		0			0
$431$	4.60788			0.221954			0.110977	−	0.993823i	$-0.464602\pi$
$431$	4.60788			0.221954			0.110977	+	0.993823i	$0.464602\pi$
$432$		0			0
$433$	8.10951			0.389718			0.194859	−	0.980831i	$-0.437575\pi$
$433$	8.10951			0.389718			0.194859	+	0.980831i	$0.437575\pi$
$434$		0			0
$435$	−33.5478	−	3.96636i	−1.60849	−	0.190172i
$436$		0			0
$437$			2.91913i			0.139641i
$438$		0			0
$439$			12.9230i			0.616780i	0.951260	+	0.308390i	$0.0997901\pi$
$439$			12.9230i			0.616780i	−0.951260	+	0.308390i	$0.900210\pi$
$440$		0			0
$441$	9.69710	+	2.32549i	0.461767	+	0.110737i
$442$		0			0
$443$	40.3652			1.91781			0.958904	−	0.283730i	$-0.0915721\pi$
$443$	40.3652			1.91781			0.958904	+	0.283730i	$0.0915721\pi$
$444$		0			0
$445$	18.7239			0.887598
$446$		0			0
$447$	−2.76708	+	23.4042i	−0.130878	+	1.10698i
$448$		0			0
$449$			38.3626i			1.81044i	0.424942	+	0.905221i	$0.360295\pi$
$449$			38.3626i			1.81044i	−0.424942	+	0.905221i	$0.639705\pi$
$450$		0			0
$451$		−	2.51624i		−	0.118485i
$452$		0			0
$453$	2.19223	−	18.5420i	0.103000	−	0.871181i
$454$		0			0
$455$	−2.13711			−0.100189
$456$		0			0
$457$	−7.34513			−0.343591			−0.171795	−	0.985133i	$-0.554957\pi$
$457$	−7.34513			−0.343591			−0.171795	+	0.985133i	$0.554957\pi$
$458$		0			0
$459$	0.745700	−	2.02366i	0.0348063	−	0.0944565i
$460$		0			0
$461$		−	13.2298i		−	0.616172i	−0.951358	−	0.308086i	$-0.900312\pi$
$461$		−	13.2298i		−	0.616172i	0.951358	−	0.308086i	$-0.0996885\pi$
$462$		0			0
$463$		−	0.124944i		−	0.00580665i	−0.999996	−	0.00290333i	$-0.999076\pi$
$463$		−	0.124944i		−	0.00580665i	0.999996	−	0.00290333i	$-0.000924159\pi$
$464$		0			0
$465$	11.3451	+	1.34134i	0.526118	+	0.0622030i
$466$		0			0
$467$	−25.5626			−1.18290			−0.591448	−	0.806343i	$-0.701444\pi$
$467$	−25.5626			−1.18290			−0.591448	+	0.806343i	$0.701444\pi$
$468$		0			0
$469$	28.4826			1.31520
$470$		0			0
$471$	17.2733	+	2.04223i	0.795914	+	0.0941010i
$472$		0			0
$473$			2.74054i			0.126010i
$474$		0			0
$475$			2.51056i			0.115192i
$476$		0			0
$477$	−6.40673	+	26.7156i	−0.293344	+	1.22322i
$478$		0			0
$479$	32.5734			1.48832			0.744158	−	0.668003i	$-0.232851\pi$
$479$	32.5734			1.48832			0.744158	+	0.668003i	$0.232851\pi$
$480$		0			0
$481$	−2.69026			−0.122665
$482$		0			0
$483$	1.13819	−	9.62691i	0.0517895	−	0.438040i
$484$		0			0
$485$			30.8651i			1.40151i
$486$		0			0
$487$			20.2162i			0.916086i	0.888930	+	0.458043i	$0.151449\pi$
$487$			20.2162i			0.916086i	−0.888930	+	0.458043i	$0.848551\pi$
$488$		0			0
$489$	−3.35197	+	28.3512i	−0.151581	+	1.28209i
$490$		0			0
$491$	−18.2908			−0.825453			−0.412726	−	0.910855i	$-0.635423\pi$
$491$	−18.2908			−0.825453			−0.412726	+	0.910855i	$0.635423\pi$
$492$		0			0
$493$	2.95383			0.133034
$494$		0			0
$495$	0.887974	−	3.70279i	0.0399115	−	0.166428i
$496$		0			0
$497$			18.6961i			0.838634i
$498$		0			0
$499$			44.6075i			1.99691i	0.0555845	+	0.998454i	$0.482298\pi$
$499$			44.6075i			1.99691i	−0.0555845	+	0.998454i	$0.517702\pi$
$500$		0			0
$501$	11.0548	+	1.30701i	0.493890	+	0.0583927i
$502$		0			0
$503$	−6.43191			−0.286785			−0.143392	−	0.989666i	$-0.545801\pi$
$503$	−6.43191			−0.286785			−0.143392	+	0.989666i	$0.545801\pi$
$504$		0			0
$505$	4.81346			0.214196
$506$		0			0
$507$	−22.0764	−	2.61009i	−0.980446	−	0.115918i
$508$		0			0
$509$			30.9858i			1.37342i	0.726930	+	0.686711i	$0.240946\pi$
$509$			30.9858i			1.37342i	−0.726930	+	0.686711i	$0.759054\pi$
$510$		0			0
$511$		−	2.18086i		−	0.0964755i
$512$		0			0
$513$	1.79664	−	4.87566i	0.0793234	−	0.215266i
$514$		0			0
$515$	−24.2694			−1.06944
$516$		0			0
$517$	−1.67599			−0.0737098
$518$		0			0
$519$	4.60288	−	38.9316i	0.202044	−	1.70891i
$520$		0			0
$521$		−	29.8712i		−	1.30868i	−0.756201	−	0.654340i	$-0.772947\pi$
$521$		−	29.8712i		−	1.30868i	0.756201	−	0.654340i	$-0.227053\pi$
$522$		0			0
$523$			33.4615i			1.46317i	0.681751	+	0.731584i	$0.261219\pi$
$523$			33.4615i			1.46317i	−0.681751	+	0.731584i	$0.738781\pi$
$524$		0			0
$525$	0.978885	−	8.27949i	0.0427220	−	0.361346i
$526$		0			0
$527$	−0.998922			−0.0435137
$528$		0			0
$529$	−14.4787			−0.629507
$530$		0			0
$531$	−23.3235	−	5.59327i	−1.01215	−	0.242727i
$532$		0			0
$533$			2.20975i			0.0957150i
$534$		0			0
$535$			16.5807i			0.716848i
$536$		0			0
$537$	29.9104	+	3.53632i	1.29073	+	0.152603i
$538$		0			0
$539$	−1.53949			−0.0663105
$540$		0			0
$541$	−23.1797			−0.996573			−0.498287	−	0.867012i	$-0.666037\pi$
$541$	−23.1797			−0.996573			−0.498287	+	0.867012i	$0.666037\pi$
$542$		0			0
$543$	−29.8231	−	3.52599i	−1.27983	−	0.151315i
$544$		0			0
$545$			26.9520i			1.15450i
$546$		0			0
$547$		−	30.3508i		−	1.29771i	−0.760913	−	0.648854i	$-0.775249\pi$
$547$		−	30.3508i		−	1.29771i	0.760913	−	0.648854i	$-0.224751\pi$
$548$		0			0
$549$	23.8009	+	5.70776i	1.01580	+	0.243601i
$550$		0			0
$551$	7.11674			0.303183
$552$		0			0
$553$	26.5248			1.12795
$554$		0			0
$555$	3.68638	−	31.1797i	0.156478	−	1.32350i
$556$		0			0
$557$		−	16.7130i		−	0.708153i	−0.935216	−	0.354077i	$-0.884795\pi$
$557$		−	16.7130i		−	0.708153i	0.935216	−	0.354077i	$-0.115205\pi$
$558$		0			0
$559$		−	2.40673i		−	0.101794i
$560$		0			0
$561$	−0.0390924	+	0.330647i	−0.00165048	+	0.0139599i
$562$		0			0
$563$	47.2888			1.99298			0.996492	−	0.0836864i	$-0.0266694\pi$
$563$	47.2888			1.99298			0.996492	+	0.0836864i	$0.0266694\pi$
$564$		0			0
$565$	34.2077			1.43913
$566$		0			0
$567$	−7.82612	+	15.3788i	−0.328666	+	0.645848i
$568$		0			0
$569$		−	7.64274i		−	0.320400i	−0.987085	−	0.160200i	$-0.948786\pi$
$569$		−	7.64274i		−	0.320400i	0.987085	−	0.160200i	$-0.0512140\pi$
$570$		0			0
$571$			5.14431i			0.215283i	0.994190	+	0.107641i	$0.0343298\pi$
$571$			5.14431i			0.215283i	−0.994190	+	0.107641i	$0.965670\pi$
$572$		0			0
$573$	10.2245	+	1.20884i	0.427134	+	0.0505001i
$574$		0			0
$575$	7.32865			0.305626
$576$		0			0
$577$	−3.84141			−0.159920			−0.0799601	−	0.996798i	$-0.525479\pi$
$577$	−3.84141			−0.159920			−0.0799601	+	0.996798i	$0.525479\pi$
$578$		0			0
$579$	7.23747	+	0.855687i	0.300779	+	0.0355611i
$580$		0			0
$581$			12.5537i			0.520816i
$582$		0			0
$583$		−	4.24130i		−	0.175657i
$584$		0			0
$585$	−0.779815	+	3.25177i	−0.0322414	+	0.134444i
$586$		0			0
$587$	−33.5811			−1.38604			−0.693020	−	0.720919i	$-0.743720\pi$
$587$	−33.5811			−1.38604			−0.693020	+	0.720919i	$0.743720\pi$
$588$		0			0
$589$	−2.40673			−0.0991675
$590$		0			0
$591$	0.261012	−	2.20766i	0.0107366	−	0.0908109i
$592$		0			0
$593$			5.85782i			0.240552i	0.992740	+	0.120276i	$0.0383780\pi$
$593$			5.85782i			0.240552i	−0.992740	+	0.120276i	$0.961622\pi$
$594$		0			0
$595$			2.18086i			0.0894065i
$596$		0			0
$597$	1.98318	−	16.7739i	0.0811661	−	0.686509i
$598$		0			0
$599$	19.5027			0.796857			0.398428	−	0.917199i	$-0.369556\pi$
$599$	19.5027			0.796857			0.398428	+	0.917199i	$0.369556\pi$
$600$		0			0
$601$	8.98747			0.366607			0.183303	−	0.983056i	$-0.441321\pi$
$601$	8.98747			0.366607			0.183303	+	0.983056i	$0.441321\pi$
$602$		0			0
$603$	10.3931	−	43.3383i	0.423238	−	1.76487i
$604$		0			0
$605$		−	29.5581i		−	1.20171i
$606$		0			0
$607$			14.8893i			0.604339i	0.953254	+	0.302170i	$0.0977109\pi$
$607$			14.8893i			0.604339i	−0.953254	+	0.302170i	$0.902289\pi$
$608$		0			0
$609$	−23.4701	−	2.77487i	−0.951055	−	0.112443i
$610$		0			0
$611$	1.47184			0.0595444
$612$		0			0
$613$	−22.3662			−0.903364			−0.451682	−	0.892179i	$-0.649176\pi$
$613$	−22.3662			−0.903364			−0.451682	+	0.892179i	$0.649176\pi$
$614$		0			0
$615$	−25.6107	−	3.02796i	−1.03272	−	0.122099i
$616$		0			0
$617$			37.1911i			1.49726i	0.662991	+	0.748628i	$0.269287\pi$
$617$			37.1911i			1.49726i	−0.662991	+	0.748628i	$0.730713\pi$
$618$		0			0
$619$		−	31.9190i		−	1.28293i	−0.767151	−	0.641467i	$-0.778326\pi$
$619$		−	31.9190i		−	1.28293i	0.767151	−	0.641467i	$-0.221674\pi$
$620$		0			0
$621$	−14.2327	−	5.24462i	−0.571139	−	0.210459i
$622$		0			0
$623$	13.0993			0.524811
$624$		0			0
$625$	−31.2499			−1.25000
$626$		0			0
$627$	−0.0941864	+	0.796636i	−0.00376144	+	0.0318146i
$628$		0			0
$629$			2.74533i			0.109463i
$630$		0			0
$631$			8.45580i			0.336620i	0.985734	+	0.168310i	$0.0538310\pi$
$631$			8.45580i			0.336620i	−0.985734	+	0.168310i	$0.946169\pi$
$632$		0			0
$633$	2.33260	−	19.7293i	0.0927126	−	0.784171i
$634$		0			0
$635$	41.1660			1.63362
$636$		0			0
$637$	1.35197			0.0535670
$638$		0			0
$639$	28.4474	+	6.82204i	1.12536	+	0.269876i
$640$		0			0
$641$		−	19.4546i		−	0.768409i	−0.923248	−	0.384205i	$-0.874476\pi$
$641$		−	19.4546i		−	0.768409i	0.923248	−	0.384205i	$-0.125524\pi$
$642$		0			0
$643$			21.9846i			0.866987i	0.901157	+	0.433493i	$0.142719\pi$
$643$			21.9846i			0.866987i	−0.901157	+	0.433493i	$0.857281\pi$
$644$		0			0
$645$	27.8936	+	3.29787i	1.09831	+	0.129853i
$646$		0			0
$647$	12.3711			0.486360			0.243180	−	0.969981i	$-0.421809\pi$
$647$	12.3711			0.486360			0.243180	+	0.969981i	$0.421809\pi$
$648$		0			0
$649$	3.70279			0.145347
$650$		0			0
$651$	7.93707	+	0.938401i	0.311078	+	0.0367788i
$652$		0			0
$653$		−	47.6312i		−	1.86395i	−0.362519	−	0.931976i	$-0.618083\pi$
$653$		−	47.6312i		−	1.86395i	0.362519	−	0.931976i	$-0.381917\pi$
$654$		0			0
$655$		−	38.9806i		−	1.52310i
$656$		0			0
$657$	−3.31833	−	0.795777i	−0.129460	−	0.0310462i
$658$		0			0
$659$	−22.1362			−0.862305			−0.431152	−	0.902279i	$-0.641893\pi$
$659$	−22.1362			−0.862305			−0.431152	+	0.902279i	$0.641893\pi$
$660$		0			0
$661$	−16.2749			−0.633022			−0.316511	−	0.948589i	$-0.602511\pi$
$661$	−16.2749			−0.633022			−0.316511	+	0.948589i	$0.602511\pi$
$662$		0			0
$663$	0.0343308	−	0.290372i	0.00133330	−	0.0112771i
$664$		0			0
$665$			5.25440i			0.203757i
$666$		0			0
$667$		−	20.7747i		−	0.804400i
$668$		0			0
$669$	−0.498031	+	4.21238i	−0.0192550	+	0.162860i
$670$		0			0
$671$	−3.77858			−0.145870
$672$		0			0
$673$	−36.3258			−1.40026			−0.700128	−	0.714018i	$-0.746874\pi$
$673$	−36.3258			−1.40026			−0.700128	+	0.714018i	$0.746874\pi$
$674$		0			0
$675$	−12.2406	−	4.51056i	−0.471142	−	0.173611i
$676$		0			0
$677$		−	6.40237i		−	0.246063i	−0.992403	−	0.123032i	$-0.960738\pi$
$677$		−	6.40237i		−	0.246063i	0.992403	−	0.123032i	$-0.0392616\pi$
$678$		0			0
$679$			21.5933i			0.828674i
$680$		0			0
$681$	−35.8277	−	4.23592i	−1.37292	−	0.162321i
$682$		0			0
$683$	−29.0891			−1.11307			−0.556533	−	0.830826i	$-0.687869\pi$
$683$	−29.0891			−1.11307			−0.556533	+	0.830826i	$0.687869\pi$
$684$		0			0
$685$	25.3874			0.970001
$686$		0			0
$687$	18.8727	+	2.23132i	0.720038	+	0.0851302i
$688$		0			0
$689$			3.72469i			0.141899i
$690$		0			0
$691$		−	28.2904i		−	1.07622i	−0.842876	−	0.538108i	$-0.819139\pi$
$691$		−	28.2904i		−	1.07622i	0.842876	−	0.538108i	$-0.180861\pi$
$692$		0			0
$693$	0.621228	−	2.59048i	0.0235985	−	0.0984040i
$694$		0			0
$695$	−52.4705			−1.99032
$696$		0			0
$697$	2.25498			0.0854136
$698$		0			0
$699$	5.03949	−	42.6244i	0.190611	−	1.61220i
$700$		0			0
$701$		−	10.1516i		−	0.383421i	−0.981452	−	0.191711i	$-0.938597\pi$
$701$		−	10.1516i		−	0.383421i	0.981452	−	0.191711i	$-0.0614035\pi$
$702$		0			0
$703$			6.61439i			0.249466i
$704$		0			0
$705$	−2.01682	+	17.0584i	−0.0759579	+	0.642458i
$706$		0			0
$707$	3.36750			0.126648
$708$		0			0
$709$	22.8557			0.858363			0.429182	−	0.903218i	$-0.358802\pi$
$709$	22.8557			0.858363			0.429182	+	0.903218i	$0.358802\pi$
$710$		0			0
$711$	9.67869	−	40.3594i	0.362979	−	1.51360i
$712$		0			0
$713$			7.02556i			0.263109i
$714$		0			0
$715$		−	0.516243i		−	0.0193064i
$716$		0			0
$717$	−43.5205	−	5.14544i	−1.62530	−	0.192160i
$718$		0			0
$719$	−31.5932			−1.17823			−0.589114	−	0.808050i	$-0.700523\pi$
$719$	−31.5932			−1.17823			−0.589114	+	0.808050i	$0.700523\pi$
$720$		0			0
$721$	−16.9789			−0.632327
$722$		0			0
$723$	23.7386	+	2.80662i	0.882847	+	0.104379i
$724$		0			0
$725$		−	17.8670i		−	0.663563i
$726$		0			0
$727$			1.51915i			0.0563420i	0.999603	+	0.0281710i	$0.00896829\pi$
$727$			1.51915i			0.0563420i	−0.999603	+	0.0281710i	$0.991032\pi$
$728$		0			0
$729$	20.5442	+	17.5196i	0.760896	+	0.648874i
$730$		0			0
$731$	−2.45599			−0.0908381
$732$		0			0
$733$	−20.5499			−0.759027			−0.379514	−	0.925186i	$-0.623909\pi$
$733$	−20.5499			−0.759027			−0.379514	+	0.925186i	$0.623909\pi$
$734$		0			0
$735$	−1.85257	+	15.6691i	−0.0683329	+	0.577965i
$736$		0			0
$737$			6.88028i			0.253438i
$738$		0			0
$739$			29.8191i			1.09691i	0.836178	+	0.548457i	$0.184785\pi$
$739$			29.8191i			1.09691i	−0.836178	+	0.548457i	$0.815215\pi$
$740$		0			0
$741$	0.0827140	−	0.699602i	0.00303857	−	0.0257005i
$742$		0			0
$743$	−37.6727			−1.38208			−0.691039	−	0.722817i	$-0.742847\pi$
$743$	−37.6727			−1.38208			−0.691039	+	0.722817i	$0.742847\pi$
$744$		0			0
$745$	−37.2892			−1.36617
$746$		0			0
$747$	19.1014	+	4.58074i	0.698882	+	0.167601i
$748$		0			0
$749$			11.5999i			0.423852i
$750$		0			0
$751$			41.4364i			1.51204i	0.654550	+	0.756018i	$0.272858\pi$
$751$			41.4364i			1.51204i	−0.654550	+	0.756018i	$0.727142\pi$
$752$		0			0
$753$	15.6523	+	1.85058i	0.570402	+	0.0674388i
$754$		0			0
$755$	29.5425			1.07516
$756$		0			0
$757$	42.8911			1.55890			0.779451	−	0.626463i	$-0.215498\pi$
$757$	42.8911			1.55890			0.779451	+	0.626463i	$0.215498\pi$
$758$		0			0
$759$	2.32549	+	0.274943i	0.0844098	+	0.00997978i
$760$		0			0
$761$		−	15.5748i		−	0.564587i	−0.959328	−	0.282293i	$-0.908905\pi$
$761$		−	15.5748i		−	0.564587i	0.959328	−	0.282293i	$-0.0910952\pi$
$762$		0			0
$763$			18.8557i			0.682622i
$764$		0			0
$765$	3.31833	+	0.795777i	0.119974	+	0.0287714i
$766$		0			0
$767$	−3.25177			−0.117415
$768$		0			0
$769$	−5.07019			−0.182836			−0.0914178	−	0.995813i	$-0.529140\pi$
$769$	−5.07019			−0.182836			−0.0914178	+	0.995813i	$0.529140\pi$
$770$		0			0
$771$	1.97329	−	16.6903i	0.0710664	−	0.601085i
$772$		0			0
$773$		−	40.2111i		−	1.44629i	−0.690694	−	0.723147i	$-0.742695\pi$
$773$		−	40.2111i		−	1.44629i	0.690694	−	0.723147i	$-0.257305\pi$
$774$		0			0
$775$			6.04223i			0.217043i
$776$		0			0
$777$	2.57900	−	21.8134i	0.0925210	−	0.782550i
$778$		0			0
$779$	5.43299			0.194657
$780$		0			0
$781$	−4.51624			−0.161604
$782$		0			0
$783$	−12.7862	+	34.6988i	−0.456941	+	1.24004i
$784$		0			0
$785$			27.5211i			0.982271i
$786$		0			0
$787$		−	30.5921i		−	1.09049i	−0.838276	−	0.545246i	$-0.816436\pi$
$787$		−	30.5921i		−	1.09049i	0.838276	−	0.545246i	$-0.183564\pi$
$788$		0			0
$789$	−32.3004	−	3.81888i	−1.14992	−	0.135956i
$790$		0			0
$791$	23.9317			0.850914
$792$		0			0
$793$	3.31833			0.117837
$794$		0			0
$795$	−43.1686	−	5.10383i	−1.53103	−	0.181014i
$796$		0			0
$797$		−	20.0472i		−	0.710108i	−0.934846	−	0.355054i	$-0.884462\pi$
$797$		−	20.0472i		−	0.710108i	0.934846	−	0.355054i	$-0.115538\pi$
$798$		0			0
$799$		−	1.50197i		−	0.0531359i
$800$		0			0
$801$	4.77981	−	19.9315i	0.168886	−	0.704244i
$802$		0			0
$803$	0.526810			0.0185907
$804$		0			0
$805$	15.3383			0.540604
$806$		0			0
$807$	1.03224	−	8.73074i	0.0363365	−	0.307337i
$808$		0			0
$809$			11.5891i			0.407452i	0.979028	+	0.203726i	$0.0653051\pi$
$809$			11.5891i			0.407452i	−0.979028	+	0.203726i	$0.934695\pi$
$810$		0			0
$811$			44.7239i			1.57047i	0.619199	+	0.785234i	$0.287458\pi$
$811$			44.7239i			1.57047i	−0.619199	+	0.785234i	$0.712542\pi$
$812$		0			0
$813$	5.72506	−	48.4230i	0.200787	−	1.69827i
$814$		0			0
$815$	−45.1712			−1.58228
$816$		0			0
$817$	−5.91729			−0.207020
$818$		0			0
$819$	−0.545560	+	2.27494i	−0.0190634	+	0.0794929i
$820$		0			0
$821$		−	30.2421i		−	1.05546i	−0.849413	−	0.527728i	$-0.823044\pi$
$821$		−	30.2421i		−	1.05546i	0.849413	−	0.527728i	$-0.176956\pi$
$822$		0			0
$823$		−	33.0901i		−	1.15345i	−0.816938	−	0.576725i	$-0.804330\pi$
$823$		−	33.0901i		−	1.15345i	0.816938	−	0.576725i	$-0.195670\pi$
$824$		0			0
$825$	2.00000	+	0.236460i	0.0696311	+	0.00823249i
$826$		0			0
$827$	−20.8332			−0.724441			−0.362221	−	0.932092i	$-0.617981\pi$
$827$	−20.8332			−0.724441			−0.362221	+	0.932092i	$0.617981\pi$
$828$		0			0
$829$	16.3594			0.568186			0.284093	−	0.958797i	$-0.408308\pi$
$829$	16.3594			0.568186			0.284093	+	0.958797i	$0.408308\pi$
$830$		0			0
$831$	17.4735	+	2.06589i	0.606149	+	0.0716650i
$832$		0			0
$833$		−	1.37964i		−	0.0478019i
$834$		0			0
$835$			17.6132i			0.609531i
$836$		0			0
$837$	4.32401	−	11.7344i	0.149460	−	0.405600i
$838$		0			0
$839$	42.9224			1.48184			0.740922	−	0.671591i	$-0.234389\pi$
$839$	42.9224			1.48184			0.740922	+	0.671591i	$0.234389\pi$
$840$		0			0
$841$	−21.6480			−0.746484
$842$		0			0
$843$	1.90065	−	16.0759i	0.0654620	−	0.553682i
$844$		0			0
$845$		−	35.1737i		−	1.21001i
$846$		0			0
$847$		−	20.6789i		−	0.710535i
$848$		0			0
$849$	−2.67853	+	22.6552i	−0.0919270	+	0.777526i
$850$		0			0
$851$	19.3083			0.661879
$852$		0			0
$853$	−8.93665			−0.305985			−0.152993	−	0.988227i	$-0.548891\pi$
$853$	−8.93665			−0.305985			−0.152993	+	0.988227i	$0.548891\pi$
$854$		0			0
$855$	7.99494	+	1.91729i	0.273421	+	0.0655698i
$856$		0			0
$857$		−	33.9961i		−	1.16129i	−0.814158	−	0.580643i	$-0.802801\pi$
$857$		−	33.9961i		−	1.16129i	0.814158	−	0.580643i	$-0.197199\pi$
$858$		0			0
$859$			31.9595i			1.09044i	0.838291	+	0.545222i	$0.183555\pi$
$859$			31.9595i			1.09044i	−0.838291	+	0.545222i	$0.816445\pi$
$860$		0			0
$861$	−17.9173	−	2.11836i	−0.610619	−	0.0721936i
$862$		0			0
$863$	−3.77777			−0.128597			−0.0642984	−	0.997931i	$-0.520481\pi$
$863$	−3.77777			−0.128597			−0.0642984	+	0.997931i	$0.520481\pi$
$864$		0			0
$865$	62.0285			2.10903
$866$		0			0
$867$	28.9449	+	3.42216i	0.983020	+	0.116223i
$868$		0			0
$869$			6.40736i			0.217355i
$870$		0			0
$871$		−	6.04223i		−	0.204733i
$872$		0			0
$873$	32.8557	+	7.87920i	1.11200	+	0.266671i
$874$		0			0
$875$	−13.0805			−0.442202
$876$		0			0
$877$	−18.8807			−0.637557			−0.318779	−	0.947829i	$-0.603273\pi$
$877$	−18.8807			−0.637557			−0.318779	+	0.947829i	$0.603273\pi$
$878$		0			0
$879$	0.785997	−	6.64803i	0.0265110	−	0.224232i
$880$		0			0
$881$			13.9442i			0.469790i	0.972021	+	0.234895i	$0.0754747\pi$
$881$			13.9442i			0.469790i	−0.972021	+	0.234895i	$0.924525\pi$
$882$		0			0
$883$			25.4210i			0.855485i	0.903901	+	0.427742i	$0.140691\pi$
$883$			25.4210i			0.855485i	−0.903901	+	0.427742i	$0.859309\pi$
$884$		0			0
$885$	4.45580	−	37.6875i	0.149780	−	1.26685i
$886$		0			0
$887$	10.0790			0.338419			0.169209	−	0.985580i	$-0.445879\pi$
$887$	10.0790			0.338419			0.169209	+	0.985580i	$0.445879\pi$
$888$		0			0
$889$	28.7998			0.965913
$890$		0			0
$891$	−3.71491	−	1.89049i	−0.124454	−	0.0633336i
$892$		0			0
$893$		−	3.61873i		−	0.121096i
$894$		0			0
$895$			47.6555i			1.59295i
$896$		0			0
$897$	−2.04223	−	0.241453i	−0.0681881	−	0.00806189i
$898$		0			0
$899$	17.1281			0.571253
$900$		0			0
$901$	3.80093			0.126627
$902$		0			0
$903$	19.5144	+	2.30719i	0.649399	+	0.0767786i
$904$		0			0
$905$		−	47.5163i		−	1.57949i
$906$		0			0
$907$		−	25.9241i		−	0.860797i	−0.902639	−	0.430398i	$-0.858373\pi$
$907$		−	25.9241i		−	0.860797i	0.902639	−	0.430398i	$-0.141627\pi$
$908$		0			0
$909$	1.22877	−	5.12389i	0.0407558	−	0.169949i
$910$		0			0
$911$	−24.2694			−0.804080			−0.402040	−	0.915622i	$-0.631699\pi$
$911$	−24.2694			−0.804080			−0.402040	+	0.915622i	$0.631699\pi$
$912$		0			0
$913$	−3.03249			−0.100361
$914$		0			0
$915$	−4.54701	+	38.4589i	−0.150319	+	1.27141i
$916$		0			0
$917$		−	27.2709i		−	0.900565i
$918$		0			0
$919$		−	18.9401i		−	0.624778i	−0.949954	−	0.312389i	$-0.898871\pi$
$919$		−	18.9401i		−	0.624778i	0.949954	−	0.312389i	$-0.101129\pi$
$920$		0			0
$921$	−0.440371	+	3.72469i	−0.0145107	+	0.122733i
$922$		0			0
$923$	3.96614			0.130547
$924$		0			0
$925$	16.6058			0.545995
$926$		0			0
$927$	−6.19545	+	25.8346i	−0.203485	+	0.848519i
$928$		0			0
$929$			18.4115i			0.604063i	0.953298	+	0.302031i	$0.0976648\pi$
$929$			18.4115i			0.604063i	−0.953298	+	0.302031i	$0.902335\pi$
$930$		0			0
$931$		−	3.32401i		−	0.108940i
$932$		0			0
$933$	−52.5839	−	6.21700i	−1.72152	−	0.203535i
$934$		0			0
$935$	−0.526810			−0.0172285
$936$		0			0
$937$	30.8066			1.00641			0.503204	−	0.864168i	$-0.332154\pi$
$937$	30.8066			1.00641			0.503204	+	0.864168i	$0.332154\pi$
$938$		0			0
$939$	41.4270	+	4.89792i	1.35192	+	0.159837i
$940$		0			0
$941$		−	48.7361i		−	1.58875i	−0.607427	−	0.794375i	$-0.707798\pi$
$941$		−	48.7361i		−	1.58875i	0.607427	−	0.794375i	$-0.292202\pi$
$942$		0			0
$943$		−	15.8596i		−	0.516460i
$944$		0			0
$945$	−25.6187	−	9.44024i	−0.833376	−	0.307091i
$946$		0			0
$947$	−51.7955			−1.68313			−0.841564	−	0.540158i	$-0.818364\pi$
$947$	−51.7955			−1.68313			−0.841564	+	0.540158i	$0.818364\pi$
$948$		0			0
$949$	−0.462642			−0.0150180
$950$		0			0
$951$	−2.03116	+	17.1797i	−0.0658648	+	0.557090i
$952$		0			0
$953$		−	11.8834i		−	0.384943i	−0.981303	−	0.192471i	$-0.938350\pi$
$953$		−	11.8834i		−	0.384943i	0.981303	−	0.192471i	$-0.0616502\pi$
$954$		0			0
$955$			16.2904i			0.527144i
$956$		0			0
$957$	0.670300	−	5.66945i	0.0216677	−	0.183267i
$958$		0			0
$959$	17.7610			0.573534
$960$		0			0
$961$	25.2077			0.813150
$962$		0			0
$963$	17.6501	+	4.23271i	0.568766	+	0.136397i
$964$		0			0
$965$			11.5313i			0.371204i
$966$		0			0
$967$			20.8807i			0.671479i	0.941955	+	0.335740i	$0.108986\pi$
$967$			20.8807i			0.671479i	−0.941955	+	0.335740i	$0.891014\pi$
$968$		0			0
$969$	−0.713922	−	0.0844071i	−0.0229345	−	0.00271155i
$970$		0			0
$971$	−49.6869			−1.59453			−0.797264	−	0.603630i	$-0.793720\pi$
$971$	−49.6869			−1.59453			−0.797264	+	0.603630i	$0.793720\pi$
$972$		0			0
$973$	−36.7085			−1.17682
$974$		0			0
$975$	−1.75639	−	0.207658i	−0.0562495	−	0.00665039i
$976$		0			0
$977$			16.2694i			0.520505i	0.965541	+	0.260253i	$0.0838058\pi$
$977$			16.2694i			0.520505i	−0.965541	+	0.260253i	$0.916194\pi$
$978$		0			0
$979$			3.16427i			0.101131i
$980$		0			0
$981$	28.6903	+	6.88028i	0.916009	+	0.219671i
$982$		0			0
$983$	34.3543			1.09573			0.547867	−	0.836566i	$-0.315440\pi$
$983$	34.3543			1.09573			0.547867	+	0.836566i	$0.315440\pi$
$984$		0			0
$985$	3.51740			0.112074
$986$		0			0
$987$	−1.41097	+	11.9341i	−0.0449117	+	0.379867i
$988$		0			0
$989$			17.2733i			0.549260i
$990$		0			0
$991$			7.13922i			0.226785i	0.993550	+	0.113392i	$0.0361717\pi$
$991$			7.13922i			0.226785i	−0.993550	+	0.113392i	$0.963828\pi$
$992$		0			0
$993$	−1.33829	+	11.3193i	−0.0424693	+	0.359209i
$994$		0			0
$995$	26.7254			0.847251
$996$		0			0
$997$	−36.4758			−1.15520			−0.577599	−	0.816320i	$-0.696010\pi$
$997$	−36.4758			−1.15520			−0.577599	+	0.816320i	$0.696010\pi$
$998$		0			0
$999$	−32.2495	−	11.8836i	−1.02033	−	0.375982i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.a.191.12	yes	12
3.2	odd	2	inner	912.2.d.a.191.2	yes	12
4.3	odd	2	inner	912.2.d.a.191.1	✓	12
12.11	even	2	inner	912.2.d.a.191.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.a.191.1	✓	12	4.3	odd	2	inner
912.2.d.a.191.2	yes	12	3.2	odd	2	inner
912.2.d.a.191.11	yes	12	12.11	even	2	inner
912.2.d.a.191.12	yes	12	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.72007 + 0.203364i) q^{3} +2.74054i q^{5} +1.91729i q^{7} +(2.91729 + 0.699602i) q^{9} -0.463141 q^{11} +0.406728 q^{13} +(-0.557328 + 4.71392i) q^{15} -0.415054i q^{17} -1.00000i q^{19} +(-0.389907 + 3.29787i) q^{21} -2.91913 q^{23} -2.51056 q^{25} +(4.87566 + 1.79664i) q^{27} +7.11674i q^{29} -2.40673i q^{31} +(-0.796636 - 0.0941864i) q^{33} -5.25440 q^{35} -6.61439 q^{37} +(0.699602 + 0.0827140i) q^{39} +5.43299i q^{41} -5.91729i q^{43} +(-1.91729 + 7.99494i) q^{45} +3.61873 q^{47} +3.32401 q^{49} +(0.0844071 - 0.713922i) q^{51} +9.15768i q^{53} -1.26926i q^{55} +(0.203364 - 1.72007i) q^{57} -7.99494 q^{59} +8.15859 q^{61} +(-1.34134 + 5.59327i) q^{63} +1.11466i q^{65} -14.8557i q^{67} +(-5.02112 - 0.593647i) q^{69} +9.75133 q^{71} -1.13747 q^{73} +(-4.31834 - 0.510558i) q^{75} -0.887974i q^{77} -13.8346i q^{79} +(8.02112 + 4.08188i) q^{81} +6.54765 q^{83} +1.13747 q^{85} +(-1.44729 + 12.2413i) q^{87} -6.83219i q^{89} +0.779815i q^{91} +(0.489442 - 4.13974i) q^{93} +2.74054 q^{95} +11.2624 q^{97} +(-1.35112 - 0.324014i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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