LMFDB - Embedded newform 1638.2.c.h.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(883,1638)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1638.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	1638.883
Dual form			1638.2.c.h.883.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.56155i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -3.56155i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.56155 q^{10} -5.56155i q^{11} +(3.56155 - 0.561553i) q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.68466 q^{17} -1.56155i q^{19} +3.56155i q^{20} -5.56155 q^{22} +6.68466 q^{23} -7.68466 q^{25} +(-0.561553 - 3.56155i) q^{26} +1.00000i q^{28} -1.56155 q^{29} -6.24621i q^{31} -1.00000i q^{32} -6.68466i q^{34} -3.56155 q^{35} +10.6847i q^{37} -1.56155 q^{38} +3.56155 q^{40} -4.00000i q^{41} -6.43845 q^{43} +5.56155i q^{44} -6.68466i q^{46} +10.2462i q^{47} -1.00000 q^{49} +7.68466i q^{50} +(-3.56155 + 0.561553i) q^{52} -4.87689 q^{53} -19.8078 q^{55} +1.00000 q^{56} +1.56155i q^{58} -4.24621i q^{59} -1.56155 q^{61} -6.24621 q^{62} -1.00000 q^{64} +(-2.00000 - 12.6847i) q^{65} -1.12311i q^{67} -6.68466 q^{68} +3.56155i q^{70} -9.36932i q^{71} +11.5616i q^{73} +10.6847 q^{74} +1.56155i q^{76} -5.56155 q^{77} +16.0000 q^{79} -3.56155i q^{80} -4.00000 q^{82} +2.00000i q^{83} -23.8078i q^{85} +6.43845i q^{86} +5.56155 q^{88} +8.00000i q^{89} +(-0.561553 - 3.56155i) q^{91} -6.68466 q^{92} +10.2462 q^{94} -5.56155 q^{95} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 6 q^{10} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 2 q^{17} - 14 q^{22} + 2 q^{23} - 6 q^{25} + 6 q^{26} + 2 q^{29} - 6 q^{35} + 2 q^{38} + 6 q^{40} - 34 q^{43} - 4 q^{49} - 6 q^{52} - 36 q^{53} - 38 q^{55} + 4 q^{56} + 2 q^{61} + 8 q^{62} - 4 q^{64} - 8 q^{65} - 2 q^{68} + 18 q^{74} - 14 q^{77} + 64 q^{79} - 16 q^{82} + 14 q^{88} + 6 q^{91} - 2 q^{92} + 8 q^{94} - 14 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 6 * q^10 + 6 * q^13 - 4 * q^14 + 4 * q^16 + 2 * q^17 - 14 * q^22 + 2 * q^23 - 6 * q^25 + 6 * q^26 + 2 * q^29 - 6 * q^35 + 2 * q^38 + 6 * q^40 - 34 * q^43 - 4 * q^49 - 6 * q^52 - 36 * q^53 - 38 * q^55 + 4 * q^56 + 2 * q^61 + 8 * q^62 - 4 * q^64 - 8 * q^65 - 2 * q^68 + 18 * q^74 - 14 * q^77 + 64 * q^79 - 16 * q^82 + 14 * q^88 + 6 * q^91 - 2 * q^92 + 8 * q^94 - 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	3.56155i		−	1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$		−	3.56155i		−	1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−3.56155			−1.12626
$11$		−	5.56155i		−	1.67687i	−0.545001	−	0.838436i	$-0.683471\pi$
$11$		−	5.56155i		−	1.67687i	0.545001	−	0.838436i	$-0.316529\pi$
$12$		0			0
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$14$	−1.00000			−0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$	6.68466			1.62127			0.810634	−	0.585553i	$-0.199123\pi$
$17$	6.68466			1.62127			0.810634	+	0.585553i	$0.199123\pi$
$18$		0			0
$19$		−	1.56155i		−	0.358245i	−0.983827	−	0.179122i	$-0.942674\pi$
$19$		−	1.56155i		−	0.358245i	0.983827	−	0.179122i	$-0.0573258\pi$
$20$			3.56155i			0.796387i
$21$		0			0
$22$	−5.56155			−1.18573
$23$	6.68466			1.39385			0.696924	−	0.717145i	$-0.254552\pi$
$23$	6.68466			1.39385			0.696924	+	0.717145i	$0.254552\pi$
$24$		0			0
$25$	−7.68466			−1.53693
$26$	−0.561553	−	3.56155i	−0.110130	−	0.698478i
$27$		0			0
$28$			1.00000i			0.188982i
$29$	−1.56155			−0.289973			−0.144987	−	0.989434i	$-0.546314\pi$
$29$	−1.56155			−0.289973			−0.144987	+	0.989434i	$0.546314\pi$
$30$		0			0
$31$		−	6.24621i		−	1.12185i	−0.827866	−	0.560926i	$-0.810445\pi$
$31$		−	6.24621i		−	1.12185i	0.827866	−	0.560926i	$-0.189555\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	6.68466i		−	1.14641i
$35$	−3.56155			−0.602012
$36$		0			0
$37$			10.6847i			1.75655i	0.478159	+	0.878274i	$0.341304\pi$
$37$			10.6847i			1.75655i	−0.478159	+	0.878274i	$0.658696\pi$
$38$	−1.56155			−0.253317
$39$		0			0
$40$	3.56155			0.563131
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	−6.43845			−0.981854			−0.490927	−	0.871201i	$-0.663342\pi$
$43$	−6.43845			−0.981854			−0.490927	+	0.871201i	$0.663342\pi$
$44$			5.56155i			0.838436i
$45$		0			0
$46$		−	6.68466i		−	0.985599i
$47$			10.2462i			1.49456i	0.664507	+	0.747282i	$0.268641\pi$
$47$			10.2462i			1.49456i	−0.664507	+	0.747282i	$0.731359\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$			7.68466i			1.08677i
$51$		0			0
$52$	−3.56155	+	0.561553i	−0.493899	+	0.0778734i
$53$	−4.87689			−0.669893			−0.334946	−	0.942237i	$-0.608718\pi$
$53$	−4.87689			−0.669893			−0.334946	+	0.942237i	$0.608718\pi$
$54$		0			0
$55$	−19.8078			−2.67088
$56$	1.00000			0.133631
$57$		0			0
$58$			1.56155i			0.205042i
$59$		−	4.24621i		−	0.552810i	−0.961041	−	0.276405i	$-0.910857\pi$
$59$		−	4.24621i		−	0.552810i	0.961041	−	0.276405i	$-0.0891431\pi$
$60$		0			0
$61$	−1.56155			−0.199936			−0.0999682	−	0.994991i	$-0.531874\pi$
$61$	−1.56155			−0.199936			−0.0999682	+	0.994991i	$0.531874\pi$
$62$	−6.24621			−0.793270
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.00000	−	12.6847i	−0.248069	−	1.57334i
$66$		0			0
$67$		−	1.12311i		−	0.137209i	−0.997644	−	0.0686046i	$-0.978145\pi$
$67$		−	1.12311i		−	0.137209i	0.997644	−	0.0686046i	$-0.0218547\pi$
$68$	−6.68466			−0.810634
$69$		0			0
$70$			3.56155i			0.425687i
$71$		−	9.36932i		−	1.11193i	−0.831205	−	0.555967i	$-0.812348\pi$
$71$		−	9.36932i		−	1.11193i	0.831205	−	0.555967i	$-0.187652\pi$
$72$		0			0
$73$			11.5616i			1.35318i	0.736361	+	0.676589i	$0.236542\pi$
$73$			11.5616i			1.35318i	−0.736361	+	0.676589i	$0.763458\pi$
$74$	10.6847			1.24207
$75$		0			0
$76$			1.56155i			0.179122i
$77$	−5.56155			−0.633798
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$		−	3.56155i		−	0.398194i
$81$		0			0
$82$	−4.00000			−0.441726
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$		−	23.8078i		−	2.58231i
$86$			6.43845i			0.694276i
$87$		0			0
$88$	5.56155			0.592864
$89$			8.00000i			0.847998i	0.905663	+	0.423999i	$0.139374\pi$
$89$			8.00000i			0.847998i	−0.905663	+	0.423999i	$0.860626\pi$
$90$		0			0
$91$	−0.561553	−	3.56155i	−0.0588667	−	0.373352i
$92$	−6.68466			−0.696924
$93$		0			0
$94$	10.2462			1.05682
$95$	−5.56155			−0.570603
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	7.68466			0.768466
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	1.80776			0.178124			0.0890621	−	0.996026i	$-0.471613\pi$
$103$	1.80776			0.178124			0.0890621	+	0.996026i	$0.471613\pi$
$104$	0.561553	+	3.56155i	0.0550648	+	0.349239i
$105$		0			0
$106$			4.87689i			0.473686i
$107$	4.87689			0.471467			0.235734	−	0.971818i	$-0.424251\pi$
$107$	4.87689			0.471467			0.235734	+	0.971818i	$0.424251\pi$
$108$		0			0
$109$			12.9309i			1.23855i	0.785173	+	0.619276i	$0.212574\pi$
$109$			12.9309i			1.23855i	−0.785173	+	0.619276i	$0.787426\pi$
$110$			19.8078i			1.88860i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	−20.2462			−1.90460			−0.952302	−	0.305158i	$-0.901291\pi$
$113$	−20.2462			−1.90460			−0.952302	+	0.305158i	$0.901291\pi$
$114$		0			0
$115$		−	23.8078i		−	2.22009i
$116$	1.56155			0.144987
$117$		0			0
$118$	−4.24621			−0.390895
$119$		−	6.68466i		−	0.612782i
$120$		0			0
$121$	−19.9309			−1.81190
$122$			1.56155i			0.141376i
$123$		0			0
$124$			6.24621i			0.560926i
$125$			9.56155i			0.855211i
$126$		0			0
$127$	10.2462			0.909204			0.454602	−	0.890695i	$-0.349781\pi$
$127$	10.2462			0.909204			0.454602	+	0.890695i	$0.349781\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−12.6847	+	2.00000i	−1.11252	+	0.175412i
$131$	1.56155			0.136434			0.0682168	−	0.997671i	$-0.478269\pi$
$131$	1.56155			0.136434			0.0682168	+	0.997671i	$0.478269\pi$
$132$		0			0
$133$	−1.56155			−0.135404
$134$	−1.12311			−0.0970215
$135$		0			0
$136$			6.68466i			0.573205i
$137$		−	6.68466i		−	0.571109i	−0.958362	−	0.285554i	$-0.907822\pi$
$137$		−	6.68466i		−	0.571109i	0.958362	−	0.285554i	$-0.0921777\pi$
$138$		0			0
$139$	10.2462			0.869072			0.434536	−	0.900654i	$-0.356912\pi$
$139$	10.2462			0.869072			0.434536	+	0.900654i	$0.356912\pi$
$140$	3.56155			0.301006
$141$		0			0
$142$	−9.36932			−0.786256
$143$	−3.12311	−	19.8078i	−0.261167	−	1.65641i
$144$		0			0
$145$			5.56155i			0.461862i
$146$	11.5616			0.956841
$147$		0			0
$148$		−	10.6847i		−	0.878274i
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$			16.6847i			1.35778i	0.734241	+	0.678889i	$0.237538\pi$
$151$			16.6847i			1.35778i	−0.734241	+	0.678889i	$0.762462\pi$
$152$	1.56155			0.126659
$153$		0			0
$154$			5.56155i			0.448163i
$155$	−22.2462			−1.78686
$156$		0			0
$157$	−11.8078			−0.942362			−0.471181	−	0.882037i	$-0.656172\pi$
$157$	−11.8078			−0.942362			−0.471181	+	0.882037i	$0.656172\pi$
$158$		−	16.0000i		−	1.27289i
$159$		0			0
$160$	−3.56155			−0.281565
$161$		−	6.68466i		−	0.526825i
$162$		0			0
$163$			9.12311i			0.714577i	0.933994	+	0.357288i	$0.116299\pi$
$163$			9.12311i			0.714577i	−0.933994	+	0.357288i	$0.883701\pi$
$164$			4.00000i			0.312348i
$165$		0			0
$166$	2.00000			0.155230
$167$			11.8078i			0.913712i	0.889541	+	0.456856i	$0.151025\pi$
$167$			11.8078i			0.913712i	−0.889541	+	0.456856i	$0.848975\pi$
$168$		0			0
$169$	12.3693	−	4.00000i	0.951486	−	0.307692i
$170$	−23.8078			−1.82597
$171$		0			0
$172$	6.43845			0.490927
$173$	3.75379			0.285395			0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379			0.285395			0.142698	+	0.989766i	$0.454422\pi$
$174$		0			0
$175$			7.68466i			0.580906i
$176$		−	5.56155i		−	0.419218i
$177$		0			0
$178$	8.00000			0.599625
$179$	−11.1231			−0.831380			−0.415690	−	0.909506i	$-0.636460\pi$
$179$	−11.1231			−0.831380			−0.415690	+	0.909506i	$0.636460\pi$
$180$		0			0
$181$	−13.3693			−0.993733			−0.496867	−	0.867827i	$-0.665516\pi$
$181$	−13.3693			−0.993733			−0.496867	+	0.867827i	$0.665516\pi$
$182$	−3.56155	+	0.561553i	−0.264000	+	0.0416251i
$183$		0			0
$184$			6.68466i			0.492800i
$185$	38.0540			2.79778
$186$		0			0
$187$		−	37.1771i		−	2.71866i
$188$		−	10.2462i		−	0.747282i
$189$		0			0
$190$			5.56155i			0.403477i
$191$	24.0540			1.74048			0.870242	−	0.492624i	$-0.163962\pi$
$191$	24.0540			1.74048			0.870242	+	0.492624i	$0.163962\pi$
$192$		0			0
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$	1.00000			0.0714286
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	4.93087			0.349540			0.174770	−	0.984609i	$-0.444082\pi$
$199$	4.93087			0.349540			0.174770	+	0.984609i	$0.444082\pi$
$200$		−	7.68466i		−	0.543387i
$201$		0			0
$202$			6.00000i			0.422159i
$203$			1.56155i			0.109600i
$204$		0			0
$205$	−14.2462			−0.994999
$206$		−	1.80776i		−	0.125953i
$207$		0			0
$208$	3.56155	−	0.561553i	0.246949	−	0.0389367i
$209$	−8.68466			−0.600730
$210$		0			0
$211$	7.80776			0.537509			0.268754	−	0.963209i	$-0.413388\pi$
$211$	7.80776			0.537509			0.268754	+	0.963209i	$0.413388\pi$
$212$	4.87689			0.334946
$213$		0			0
$214$		−	4.87689i		−	0.333378i
$215$			22.9309i			1.56387i
$216$		0			0
$217$	−6.24621			−0.424020
$218$	12.9309			0.875789
$219$		0			0
$220$	19.8078			1.33544
$221$	23.8078	−	3.75379i	1.60148	−	0.252507i
$222$		0			0
$223$		−	1.75379i		−	0.117442i	−0.998274	−	0.0587212i	$-0.981298\pi$
$223$		−	1.75379i		−	0.117442i	0.998274	−	0.0587212i	$-0.0187023\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$			20.2462i			1.34676i
$227$			17.6155i			1.16918i	0.811328	+	0.584592i	$0.198745\pi$
$227$			17.6155i			1.16918i	−0.811328	+	0.584592i	$0.801255\pi$
$228$		0			0
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	−23.8078			−1.56984
$231$		0			0
$232$		−	1.56155i		−	0.102521i
$233$	−2.00000			−0.131024			−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000			−0.131024			−0.0655122	+	0.997852i	$0.520868\pi$
$234$		0			0
$235$	36.4924			2.38050
$236$			4.24621i			0.276405i
$237$		0			0
$238$	−6.68466			−0.433302
$239$		−	14.2462i		−	0.921511i	−0.887527	−	0.460755i	$-0.847579\pi$
$239$		−	14.2462i		−	0.921511i	0.887527	−	0.460755i	$-0.152421\pi$
$240$		0			0
$241$			6.00000i			0.386494i	0.981150	+	0.193247i	$0.0619019\pi$
$241$			6.00000i			0.386494i	−0.981150	+	0.193247i	$0.938098\pi$
$242$			19.9309i			1.28120i
$243$		0			0
$244$	1.56155			0.0999682
$245$			3.56155i			0.227539i
$246$		0			0
$247$	−0.876894	−	5.56155i	−0.0557955	−	0.353873i
$248$	6.24621			0.396635
$249$		0			0
$250$	9.56155			0.604726
$251$	−22.0540			−1.39203			−0.696017	−	0.718025i	$-0.745046\pi$
$251$	−22.0540			−1.39203			−0.696017	+	0.718025i	$0.745046\pi$
$252$		0			0
$253$		−	37.1771i		−	2.33730i
$254$		−	10.2462i		−	0.642904i
$255$		0			0
$256$	1.00000			0.0625000
$257$	26.0000			1.62184			0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000			1.62184			0.810918	+	0.585160i	$0.198968\pi$
$258$		0			0
$259$	10.6847			0.663912
$260$	2.00000	+	12.6847i	0.124035	+	0.786669i
$261$		0			0
$262$		−	1.56155i		−	0.0964731i
$263$	23.3693			1.44101			0.720507	−	0.693448i	$-0.243909\pi$
$263$	23.3693			1.44101			0.720507	+	0.693448i	$0.243909\pi$
$264$		0			0
$265$			17.3693i			1.06699i
$266$			1.56155i			0.0957449i
$267$		0			0
$268$			1.12311i			0.0686046i
$269$	−31.3693			−1.91262			−0.956311	−	0.292353i	$-0.905562\pi$
$269$	−31.3693			−1.91262			−0.956311	+	0.292353i	$0.905562\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$	6.68466			0.405317
$273$		0			0
$274$	−6.68466			−0.403835
$275$			42.7386i			2.57724i
$276$		0			0
$277$	−10.8769			−0.653529			−0.326765	−	0.945106i	$-0.605958\pi$
$277$	−10.8769			−0.653529			−0.326765	+	0.945106i	$0.605958\pi$
$278$		−	10.2462i		−	0.614527i
$279$		0			0
$280$		−	3.56155i		−	0.212843i
$281$		−	6.00000i		−	0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$		−	6.00000i		−	0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$		0			0
$283$	−4.87689			−0.289901			−0.144951	−	0.989439i	$-0.546302\pi$
$283$	−4.87689			−0.289901			−0.144951	+	0.989439i	$0.546302\pi$
$284$			9.36932i			0.555967i
$285$		0			0
$286$	−19.8078	+	3.12311i	−1.17126	+	0.184673i
$287$	−4.00000			−0.236113
$288$		0			0
$289$	27.6847			1.62851
$290$	5.56155			0.326586
$291$		0			0
$292$		−	11.5616i		−	0.676589i
$293$			7.75379i			0.452981i	0.974013	+	0.226491i	$0.0727253\pi$
$293$			7.75379i			0.452981i	−0.974013	+	0.226491i	$0.927275\pi$
$294$		0			0
$295$	−15.1231			−0.880501
$296$	−10.6847			−0.621033
$297$		0			0
$298$	−10.0000			−0.579284
$299$	23.8078	−	3.75379i	1.37684	−	0.217087i
$300$		0			0
$301$			6.43845i			0.371106i
$302$	16.6847			0.960094
$303$		0			0
$304$		−	1.56155i		−	0.0895612i
$305$			5.56155i			0.318454i
$306$		0			0
$307$			26.2462i			1.49795i	0.662598	+	0.748975i	$0.269454\pi$
$307$			26.2462i			1.49795i	−0.662598	+	0.748975i	$0.730546\pi$
$308$	5.56155			0.316899
$309$		0			0
$310$			22.2462i			1.26350i
$311$	0.492423			0.0279227			0.0139614	−	0.999903i	$-0.495556\pi$
$311$	0.492423			0.0279227			0.0139614	+	0.999903i	$0.495556\pi$
$312$		0			0
$313$	32.2462			1.82266			0.911332	−	0.411673i	$-0.135055\pi$
$313$	32.2462			1.82266			0.911332	+	0.411673i	$0.135055\pi$
$314$			11.8078i			0.666351i
$315$		0			0
$316$	−16.0000			−0.900070
$317$			3.36932i			0.189240i	0.995513	+	0.0946198i	$0.0301636\pi$
$317$			3.36932i			0.189240i	−0.995513	+	0.0946198i	$0.969836\pi$
$318$		0			0
$319$			8.68466i			0.486248i
$320$			3.56155i			0.199097i
$321$		0			0
$322$	−6.68466			−0.372521
$323$		−	10.4384i		−	0.580811i
$324$		0			0
$325$	−27.3693	+	4.31534i	−1.51818	+	0.239372i
$326$	9.12311			0.505282
$327$		0			0
$328$	4.00000			0.220863
$329$	10.2462			0.564892
$330$		0			0
$331$		−	1.12311i		−	0.0617315i	−0.999524	−	0.0308657i	$-0.990174\pi$
$331$		−	1.12311i		−	0.0617315i	0.999524	−	0.0308657i	$-0.00982643\pi$
$332$		−	2.00000i		−	0.109764i
$333$		0			0
$334$	11.8078			0.646092
$335$	−4.00000			−0.218543
$336$		0			0
$337$	20.0540			1.09241			0.546205	−	0.837652i	$-0.316072\pi$
$337$	20.0540			1.09241			0.546205	+	0.837652i	$0.316072\pi$
$338$	−4.00000	−	12.3693i	−0.217571	−	0.672802i
$339$		0			0
$340$			23.8078i			1.29116i
$341$	−34.7386			−1.88120
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		−	6.43845i		−	0.347138i
$345$		0			0
$346$		−	3.75379i		−	0.201805i
$347$	24.4924			1.31482			0.657411	−	0.753532i	$-0.271652\pi$
$347$	24.4924			1.31482			0.657411	+	0.753532i	$0.271652\pi$
$348$		0			0
$349$			3.36932i			0.180355i	0.995926	+	0.0901777i	$0.0287435\pi$
$349$			3.36932i			0.180355i	−0.995926	+	0.0901777i	$0.971256\pi$
$350$	7.68466			0.410762
$351$		0			0
$352$	−5.56155			−0.296432
$353$			18.2462i			0.971148i	0.874196	+	0.485574i	$0.161389\pi$
$353$			18.2462i			0.971148i	−0.874196	+	0.485574i	$0.838611\pi$
$354$		0			0
$355$	−33.3693			−1.77106
$356$		−	8.00000i		−	0.423999i
$357$		0			0
$358$			11.1231i			0.587874i
$359$		−	23.6155i		−	1.24638i	−0.782071	−	0.623190i	$-0.785836\pi$
$359$		−	23.6155i		−	1.24638i	0.782071	−	0.623190i	$-0.214164\pi$
$360$		0			0
$361$	16.5616			0.871661
$362$			13.3693i			0.702676i
$363$		0			0
$364$	0.561553	+	3.56155i	0.0294334	+	0.186676i
$365$	41.1771			2.15531
$366$		0			0
$367$	0.630683			0.0329214			0.0164607	−	0.999865i	$-0.494760\pi$
$367$	0.630683			0.0329214			0.0164607	+	0.999865i	$0.494760\pi$
$368$	6.68466			0.348462
$369$		0			0
$370$		−	38.0540i		−	1.97833i
$371$			4.87689i			0.253196i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$	−37.1771			−1.92238
$375$		0			0
$376$	−10.2462			−0.528408
$377$	−5.56155	+	0.876894i	−0.286435	+	0.0451624i
$378$		0			0
$379$		−	18.8769i		−	0.969641i	−0.874614	−	0.484820i	$-0.838885\pi$
$379$		−	18.8769i		−	0.969641i	0.874614	−	0.484820i	$-0.161115\pi$
$380$	5.56155			0.285302
$381$		0			0
$382$		−	24.0540i		−	1.23071i
$383$		−	17.5616i		−	0.897353i	−0.893694	−	0.448677i	$-0.851895\pi$
$383$		−	17.5616i		−	0.897353i	0.893694	−	0.448677i	$-0.148105\pi$
$384$		0			0
$385$			19.8078i			1.00950i
$386$	−12.0000			−0.610784
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	−27.1231			−1.37520			−0.687598	−	0.726092i	$-0.741335\pi$
$389$	−27.1231			−1.37520			−0.687598	+	0.726092i	$0.741335\pi$
$390$		0			0
$391$	44.6847			2.25980
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	6.00000			0.302276
$395$		−	56.9848i		−	2.86722i
$396$		0			0
$397$			15.3693i			0.771364i	0.922632	+	0.385682i	$0.126034\pi$
$397$			15.3693i			0.771364i	−0.922632	+	0.385682i	$0.873966\pi$
$398$		−	4.93087i		−	0.247162i
$399$		0			0
$400$	−7.68466			−0.384233
$401$			30.9848i			1.54731i	0.633608	+	0.773655i	$0.281573\pi$
$401$			30.9848i			1.54731i	−0.633608	+	0.773655i	$0.718427\pi$
$402$		0			0
$403$	−3.50758	−	22.2462i	−0.174725	−	1.10816i
$404$	6.00000			0.298511
$405$		0			0
$406$	1.56155			0.0774986
$407$	59.4233			2.94550
$408$		0			0
$409$			13.8078i			0.682750i	0.939927	+	0.341375i	$0.110893\pi$
$409$			13.8078i			0.682750i	−0.939927	+	0.341375i	$0.889107\pi$
$410$			14.2462i			0.703570i
$411$		0			0
$412$	−1.80776			−0.0890621
$413$	−4.24621			−0.208942
$414$		0			0
$415$	7.12311			0.349660
$416$	−0.561553	−	3.56155i	−0.0275324	−	0.174619i
$417$		0			0
$418$			8.68466i			0.424781i
$419$	20.6847			1.01051			0.505256	−	0.862970i	$-0.331398\pi$
$419$	20.6847			1.01051			0.505256	+	0.862970i	$0.331398\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$		−	7.80776i		−	0.380076i
$423$		0			0
$424$		−	4.87689i		−	0.236843i
$425$	−51.3693			−2.49178
$426$		0			0
$427$			1.56155i			0.0755688i
$428$	−4.87689			−0.235734
$429$		0			0
$430$	22.9309			1.10582
$431$		−	29.8617i		−	1.43839i	−0.694809	−	0.719195i	$-0.744511\pi$
$431$		−	29.8617i		−	1.43839i	0.694809	−	0.719195i	$-0.255489\pi$
$432$		0			0
$433$	23.3693			1.12306			0.561529	−	0.827457i	$-0.310213\pi$
$433$	23.3693			1.12306			0.561529	+	0.827457i	$0.310213\pi$
$434$			6.24621i			0.299828i
$435$		0			0
$436$		−	12.9309i		−	0.619276i
$437$		−	10.4384i		−	0.499339i
$438$		0			0
$439$	−8.93087			−0.426247			−0.213124	−	0.977025i	$-0.568364\pi$
$439$	−8.93087			−0.426247			−0.213124	+	0.977025i	$0.568364\pi$
$440$		−	19.8078i		−	0.944298i
$441$		0			0
$442$	−3.75379	−	23.8078i	−0.178550	−	1.13242i
$443$	−6.63068			−0.315033			−0.157517	−	0.987516i	$-0.550349\pi$
$443$	−6.63068			−0.315033			−0.157517	+	0.987516i	$0.550349\pi$
$444$		0			0
$445$	28.4924			1.35067
$446$	−1.75379			−0.0830443
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			9.31534i			0.439618i	0.975543	+	0.219809i	$0.0705434\pi$
$449$			9.31534i			0.439618i	−0.975543	+	0.219809i	$0.929457\pi$
$450$		0			0
$451$	−22.2462			−1.04753
$452$	20.2462			0.952302
$453$		0			0
$454$	17.6155			0.826738
$455$	−12.6847	+	2.00000i	−0.594666	+	0.0937614i
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$	2.00000			0.0934539
$459$		0			0
$460$			23.8078i			1.11004i
$461$		−	36.9309i		−	1.72004i	−0.510259	−	0.860021i	$-0.670450\pi$
$461$		−	36.9309i		−	1.72004i	0.510259	−	0.860021i	$-0.329550\pi$
$462$		0			0
$463$		−	30.5464i		−	1.41961i	−0.704397	−	0.709806i	$-0.748783\pi$
$463$		−	30.5464i		−	1.41961i	0.704397	−	0.709806i	$-0.251217\pi$
$464$	−1.56155			−0.0724933
$465$		0			0
$466$			2.00000i			0.0926482i
$467$	−1.56155			−0.0722600			−0.0361300	−	0.999347i	$-0.511503\pi$
$467$	−1.56155			−0.0722600			−0.0361300	+	0.999347i	$0.511503\pi$
$468$		0			0
$469$	−1.12311			−0.0518602
$470$		−	36.4924i		−	1.68327i
$471$		0			0
$472$	4.24621			0.195448
$473$			35.8078i			1.64644i
$474$		0			0
$475$			12.0000i			0.550598i
$476$			6.68466i			0.306391i
$477$		0			0
$478$	−14.2462			−0.651607
$479$		−	12.3002i		−	0.562010i	−0.959706	−	0.281005i	$-0.909332\pi$
$479$		−	12.3002i		−	0.562010i	0.959706	−	0.281005i	$-0.0906677\pi$
$480$		0			0
$481$	6.00000	+	38.0540i	0.273576	+	1.73511i
$482$	6.00000			0.273293
$483$		0			0
$484$	19.9309			0.905949
$485$	35.6155			1.61722
$486$		0			0
$487$		−	13.7538i		−	0.623244i	−0.950206	−	0.311622i	$-0.899128\pi$
$487$		−	13.7538i		−	0.623244i	0.950206	−	0.311622i	$-0.100872\pi$
$488$		−	1.56155i		−	0.0706882i
$489$		0			0
$490$	3.56155			0.160895
$491$	10.2462			0.462405			0.231203	−	0.972906i	$-0.425734\pi$
$491$	10.2462			0.462405			0.231203	+	0.972906i	$0.425734\pi$
$492$		0			0
$493$	−10.4384			−0.470124
$494$	−5.56155	+	0.876894i	−0.250226	+	0.0394533i
$495$		0			0
$496$		−	6.24621i		−	0.280463i
$497$	−9.36932			−0.420271
$498$		0			0
$499$		−	1.50758i		−	0.0674884i	−0.999431	−	0.0337442i	$-0.989257\pi$
$499$		−	1.50758i		−	0.0674884i	0.999431	−	0.0337442i	$-0.0107432\pi$
$500$		−	9.56155i		−	0.427606i
$501$		0			0
$502$			22.0540i			0.984317i
$503$	−35.6155			−1.58802			−0.794009	−	0.607906i	$-0.792010\pi$
$503$	−35.6155			−1.58802			−0.794009	+	0.607906i	$0.792010\pi$
$504$		0			0
$505$			21.3693i			0.950922i
$506$	−37.1771			−1.65272
$507$		0			0
$508$	−10.2462			−0.454602
$509$		−	2.68466i		−	0.118995i	−0.998228	−	0.0594977i	$-0.981050\pi$
$509$		−	2.68466i		−	0.118995i	0.998228	−	0.0594977i	$-0.0189499\pi$
$510$		0			0
$511$	11.5616			0.511453
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	26.0000i		−	1.14681i
$515$		−	6.43845i		−	0.283712i
$516$		0			0
$517$	56.9848			2.50619
$518$		−	10.6847i		−	0.469457i
$519$		0			0
$520$	12.6847	−	2.00000i	0.556259	−	0.0877058i
$521$	−8.05398			−0.352851			−0.176426	−	0.984314i	$-0.556453\pi$
$521$	−8.05398			−0.352851			−0.176426	+	0.984314i	$0.556453\pi$
$522$		0			0
$523$	−8.49242			−0.371348			−0.185674	−	0.982611i	$-0.559447\pi$
$523$	−8.49242			−0.371348			−0.185674	+	0.982611i	$0.559447\pi$
$524$	−1.56155			−0.0682168
$525$		0			0
$526$		−	23.3693i		−	1.01895i
$527$		−	41.7538i		−	1.81882i
$528$		0			0
$529$	21.6847			0.942811
$530$	17.3693			0.754475
$531$		0			0
$532$	1.56155			0.0677019
$533$	−2.24621	−	14.2462i	−0.0972942	−	0.617072i
$534$		0			0
$535$		−	17.3693i		−	0.750941i
$536$	1.12311			0.0485108
$537$		0			0
$538$			31.3693i			1.35243i
$539$			5.56155i			0.239553i
$540$		0			0
$541$		−	6.19224i		−	0.266225i	−0.991101	−	0.133113i	$-0.957503\pi$
$541$		−	6.19224i		−	0.266225i	0.991101	−	0.133113i	$-0.0424972\pi$
$542$	−8.00000			−0.343629
$543$		0			0
$544$		−	6.68466i		−	0.286602i
$545$	46.0540			1.97274
$546$		0			0
$547$	28.9848			1.23930			0.619651	−	0.784877i	$-0.287274\pi$
$547$	28.9848			1.23930			0.619651	+	0.784877i	$0.287274\pi$
$548$			6.68466i			0.285554i
$549$		0			0
$550$	42.7386			1.82238
$551$			2.43845i			0.103881i
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$			10.8769i			0.462115i
$555$		0			0
$556$	−10.2462			−0.434536
$557$		−	16.2462i		−	0.688374i	−0.938901	−	0.344187i	$-0.888155\pi$
$557$		−	16.2462i		−	0.688374i	0.938901	−	0.344187i	$-0.111845\pi$
$558$		0			0
$559$	−22.9309	+	3.61553i	−0.969872	+	0.152921i
$560$	−3.56155			−0.150503
$561$		0			0
$562$	−6.00000			−0.253095
$563$	−38.0540			−1.60378			−0.801892	−	0.597469i	$-0.796173\pi$
$563$	−38.0540			−1.60378			−0.801892	+	0.597469i	$0.796173\pi$
$564$		0			0
$565$			72.1080i			3.03360i
$566$			4.87689i			0.204991i
$567$		0			0
$568$	9.36932			0.393128
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	−18.2462			−0.763580			−0.381790	−	0.924249i	$-0.624692\pi$
$571$	−18.2462			−0.763580			−0.381790	+	0.924249i	$0.624692\pi$
$572$	3.12311	+	19.8078i	0.130584	+	0.828204i
$573$		0			0
$574$			4.00000i			0.166957i
$575$	−51.3693			−2.14225
$576$		0			0
$577$			7.75379i			0.322794i	0.986890	+	0.161397i	$0.0516000\pi$
$577$			7.75379i			0.322794i	−0.986890	+	0.161397i	$0.948400\pi$
$578$		−	27.6847i		−	1.15153i
$579$		0			0
$580$		−	5.56155i		−	0.230931i
$581$	2.00000			0.0829740
$582$		0			0
$583$			27.1231i			1.12332i
$584$	−11.5616			−0.478420
$585$		0			0
$586$	7.75379			0.320306
$587$			37.1231i			1.53223i	0.642701	+	0.766117i	$0.277814\pi$
$587$			37.1231i			1.53223i	−0.642701	+	0.766117i	$0.722186\pi$
$588$		0			0
$589$	−9.75379			−0.401898
$590$			15.1231i			0.622608i
$591$		0			0
$592$			10.6847i			0.439137i
$593$			28.0000i			1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$			28.0000i			1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$		0			0
$595$	−23.8078			−0.976023
$596$			10.0000i			0.409616i
$597$		0			0
$598$	−3.75379	−	23.8078i	−0.153504	−	0.973572i
$599$	−30.3002			−1.23803			−0.619016	−	0.785378i	$-0.712468\pi$
$599$	−30.3002			−1.23803			−0.619016	+	0.785378i	$0.712468\pi$
$600$		0			0
$601$	−31.8617			−1.29967			−0.649834	−	0.760076i	$-0.725161\pi$
$601$	−31.8617			−1.29967			−0.649834	+	0.760076i	$0.725161\pi$
$602$	6.43845			0.262412
$603$		0			0
$604$		−	16.6847i		−	0.678889i
$605$			70.9848i			2.88594i
$606$		0			0
$607$	−28.5464			−1.15866			−0.579331	−	0.815092i	$-0.696686\pi$
$607$	−28.5464			−1.15866			−0.579331	+	0.815092i	$0.696686\pi$
$608$	−1.56155			−0.0633293
$609$		0			0
$610$	5.56155			0.225181
$611$	5.75379	+	36.4924i	0.232773	+	1.47633i
$612$		0			0
$613$		−	1.80776i		−	0.0730149i	−0.999333	−	0.0365075i	$-0.988377\pi$
$613$		−	1.80776i		−	0.0730149i	0.999333	−	0.0365075i	$-0.0116233\pi$
$614$	26.2462			1.05921
$615$		0			0
$616$		−	5.56155i		−	0.224081i
$617$		−	6.19224i		−	0.249290i	−0.992201	−	0.124645i	$-0.960221\pi$
$617$		−	6.19224i		−	0.249290i	0.992201	−	0.124645i	$-0.0397792\pi$
$618$		0			0
$619$		−	30.9309i		−	1.24322i	−0.783328	−	0.621608i	$-0.786480\pi$
$619$		−	30.9309i		−	1.24322i	0.783328	−	0.621608i	$-0.213520\pi$
$620$	22.2462			0.893429
$621$		0			0
$622$		−	0.492423i		−	0.0197443i
$623$	8.00000			0.320513
$624$		0			0
$625$	−4.36932			−0.174773
$626$		−	32.2462i		−	1.28882i
$627$		0			0
$628$	11.8078			0.471181
$629$			71.4233i			2.84783i
$630$		0			0
$631$			12.6847i			0.504968i	0.967601	+	0.252484i	$0.0812476\pi$
$631$			12.6847i			0.504968i	−0.967601	+	0.252484i	$0.918752\pi$
$632$			16.0000i			0.636446i
$633$		0			0
$634$	3.36932			0.133813
$635$		−	36.4924i		−	1.44816i
$636$		0			0
$637$	−3.56155	+	0.561553i	−0.141114	+	0.0222495i
$638$	8.68466			0.343829
$639$		0			0
$640$	3.56155			0.140783
$641$	−30.1080			−1.18919			−0.594596	−	0.804024i	$-0.702688\pi$
$641$	−30.1080			−1.18919			−0.594596	+	0.804024i	$0.702688\pi$
$642$		0			0
$643$		−	0.192236i		−	0.00758105i	−0.999993	−	0.00379052i	$-0.998793\pi$
$643$		−	0.192236i		−	0.00758105i	0.999993	−	0.00379052i	$-0.00120656\pi$
$644$			6.68466i			0.263412i
$645$		0			0
$646$	−10.4384			−0.410695
$647$	−10.6307			−0.417935			−0.208968	−	0.977923i	$-0.567010\pi$
$647$	−10.6307			−0.417935			−0.208968	+	0.977923i	$0.567010\pi$
$648$		0			0
$649$	−23.6155			−0.926991
$650$	4.31534	+	27.3693i	0.169262	+	1.07351i
$651$		0			0
$652$		−	9.12311i		−	0.357288i
$653$	29.5616			1.15683			0.578416	−	0.815742i	$-0.303671\pi$
$653$	29.5616			1.15683			0.578416	+	0.815742i	$0.303671\pi$
$654$		0			0
$655$		−	5.56155i		−	0.217308i
$656$		−	4.00000i		−	0.156174i
$657$		0			0
$658$		−	10.2462i		−	0.399439i
$659$	−12.8769			−0.501613			−0.250806	−	0.968037i	$-0.580696\pi$
$659$	−12.8769			−0.501613			−0.250806	+	0.968037i	$0.580696\pi$
$660$		0			0
$661$		−	44.7386i		−	1.74013i	−0.492936	−	0.870066i	$-0.664076\pi$
$661$		−	44.7386i		−	1.74013i	0.492936	−	0.870066i	$-0.335924\pi$
$662$	−1.12311			−0.0436507
$663$		0			0
$664$	−2.00000			−0.0776151
$665$			5.56155i			0.215668i
$666$		0			0
$667$	−10.4384			−0.404178
$668$		−	11.8078i		−	0.456856i
$669$		0			0
$670$			4.00000i			0.154533i
$671$			8.68466i			0.335268i
$672$		0			0
$673$	37.8078			1.45738			0.728691	−	0.684843i	$-0.240129\pi$
$673$	37.8078			1.45738			0.728691	+	0.684843i	$0.240129\pi$
$674$		−	20.0540i		−	0.772450i
$675$		0			0
$676$	−12.3693	+	4.00000i	−0.475743	+	0.153846i
$677$	−31.3693			−1.20562			−0.602810	−	0.797884i	$-0.705952\pi$
$677$	−31.3693			−1.20562			−0.602810	+	0.797884i	$0.705952\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$	23.8078			0.912986
$681$		0			0
$682$			34.7386i			1.33021i
$683$		−	0.684658i		−	0.0261977i	−0.999914	−	0.0130989i	$-0.995830\pi$
$683$		−	0.684658i		−	0.0261977i	0.999914	−	0.0130989i	$-0.00416962\pi$
$684$		0			0
$685$	−23.8078			−0.909648
$686$	1.00000			0.0381802
$687$		0			0
$688$	−6.43845			−0.245463
$689$	−17.3693	+	2.73863i	−0.661718	+	0.104334i
$690$		0			0
$691$			36.4924i			1.38824i	0.719861	+	0.694119i	$0.244206\pi$
$691$			36.4924i			1.38824i	−0.719861	+	0.694119i	$0.755794\pi$
$692$	−3.75379			−0.142698
$693$		0			0
$694$		−	24.4924i		−	0.929720i
$695$		−	36.4924i		−	1.38424i
$696$		0			0
$697$		−	26.7386i		−	1.01280i
$698$	3.36932			0.127531
$699$		0			0
$700$		−	7.68466i		−	0.290453i
$701$	3.61553			0.136557			0.0682783	−	0.997666i	$-0.478249\pi$
$701$	3.61553			0.136557			0.0682783	+	0.997666i	$0.478249\pi$
$702$		0			0
$703$	16.6847			0.629274
$704$			5.56155i			0.209609i
$705$		0			0
$706$	18.2462			0.686705
$707$			6.00000i			0.225653i
$708$		0			0
$709$			31.7538i			1.19254i	0.802784	+	0.596269i	$0.203351\pi$
$709$			31.7538i			1.19254i	−0.802784	+	0.596269i	$0.796649\pi$
$710$			33.3693i			1.25233i
$711$		0			0
$712$	−8.00000			−0.299813
$713$		−	41.7538i		−	1.56369i
$714$		0			0
$715$	−70.5464	+	11.1231i	−2.63829	+	0.415981i
$716$	11.1231			0.415690
$717$		0			0
$718$	−23.6155			−0.881324
$719$	−13.7538			−0.512930			−0.256465	−	0.966554i	$-0.582558\pi$
$719$	−13.7538			−0.512930			−0.256465	+	0.966554i	$0.582558\pi$
$720$		0			0
$721$		−	1.80776i		−	0.0673247i
$722$		−	16.5616i		−	0.616357i
$723$		0			0
$724$	13.3693			0.496867
$725$	12.0000			0.445669
$726$		0			0
$727$	−48.9309			−1.81475			−0.907373	−	0.420327i	$-0.861915\pi$
$727$	−48.9309			−1.81475			−0.907373	+	0.420327i	$0.861915\pi$
$728$	3.56155	−	0.561553i	0.132000	−	0.0208125i
$729$		0			0
$730$		−	41.1771i		−	1.52403i
$731$	−43.0388			−1.59185
$732$		0			0
$733$		−	22.8769i		−	0.844977i	−0.906368	−	0.422489i	$-0.861157\pi$
$733$		−	22.8769i		−	0.844977i	0.906368	−	0.422489i	$-0.138843\pi$
$734$		−	0.630683i		−	0.0232789i
$735$		0			0
$736$		−	6.68466i		−	0.246400i
$737$	−6.24621			−0.230082
$738$		0			0
$739$		−	17.6155i		−	0.647998i	−0.946057	−	0.323999i	$-0.894973\pi$
$739$		−	17.6155i		−	0.647998i	0.946057	−	0.323999i	$-0.105027\pi$
$740$	−38.0540			−1.39889
$741$		0			0
$742$	4.87689			0.179036
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−35.6155			−1.30485
$746$			6.00000i			0.219676i
$747$		0			0
$748$			37.1771i			1.35933i
$749$		−	4.87689i		−	0.178198i
$750$		0			0
$751$	2.24621			0.0819654			0.0409827	−	0.999160i	$-0.486951\pi$
$751$	2.24621			0.0819654			0.0409827	+	0.999160i	$0.486951\pi$
$752$			10.2462i			0.373641i
$753$		0			0
$754$	0.876894	+	5.56155i	0.0319346	+	0.202540i
$755$	59.4233			2.16264
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$	−18.8769			−0.685640
$759$		0			0
$760$		−	5.56155i		−	0.201739i
$761$			39.1231i			1.41821i	0.705102	+	0.709106i	$0.250901\pi$
$761$			39.1231i			1.41821i	−0.705102	+	0.709106i	$0.749099\pi$
$762$		0			0
$763$	12.9309			0.468129
$764$	−24.0540			−0.870242
$765$		0			0
$766$	−17.5616			−0.634525
$767$	−2.38447	−	15.1231i	−0.0860983	−	0.546064i
$768$		0			0
$769$		−	8.43845i		−	0.304298i	−0.988358	−	0.152149i	$-0.951381\pi$
$769$		−	8.43845i		−	0.304298i	0.988358	−	0.152149i	$-0.0486194\pi$
$770$	19.8078			0.713822
$771$		0			0
$772$			12.0000i			0.431889i
$773$		−	12.9309i		−	0.465091i	−0.972586	−	0.232546i	$-0.925295\pi$
$773$		−	12.9309i		−	0.465091i	0.972586	−	0.232546i	$-0.0747055\pi$
$774$		0			0
$775$			48.0000i			1.72421i
$776$	−10.0000			−0.358979
$777$		0			0
$778$			27.1231i			0.972410i
$779$	−6.24621			−0.223794
$780$		0			0
$781$	−52.1080			−1.86457
$782$		−	44.6847i		−	1.59792i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			42.0540i			1.50097i
$786$		0			0
$787$			17.0691i			0.608449i	0.952600	+	0.304224i	$0.0983973\pi$
$787$			17.0691i			0.608449i	−0.952600	+	0.304224i	$0.901603\pi$
$788$		−	6.00000i		−	0.213741i
$789$		0			0
$790$	−56.9848			−2.02743
$791$			20.2462i			0.719872i
$792$		0			0
$793$	−5.56155	+	0.876894i	−0.197497	+	0.0311394i
$794$	15.3693			0.545437
$795$		0			0
$796$	−4.93087			−0.174770
$797$	−3.75379			−0.132966			−0.0664830	−	0.997788i	$-0.521178\pi$
$797$	−3.75379			−0.132966			−0.0664830	+	0.997788i	$0.521178\pi$
$798$		0			0
$799$			68.4924i			2.42309i
$800$			7.68466i			0.271694i
$801$		0			0
$802$	30.9848			1.09411
$803$	64.3002			2.26910
$804$		0			0
$805$	−23.8078			−0.839113
$806$	−22.2462	+	3.50758i	−0.783589	+	0.123549i
$807$		0			0
$808$		−	6.00000i		−	0.211079i
$809$	55.3693			1.94668			0.973341	−	0.229364i	$-0.0736646\pi$
$809$	55.3693			1.94668			0.973341	+	0.229364i	$0.0736646\pi$
$810$		0			0
$811$		−	2.05398i		−	0.0721248i	−0.999350	−	0.0360624i	$-0.988518\pi$
$811$		−	2.05398i		−	0.0721248i	0.999350	−	0.0360624i	$-0.0114815\pi$
$812$		−	1.56155i		−	0.0547998i
$813$		0			0
$814$		−	59.4233i		−	2.08279i
$815$	32.4924			1.13816
$816$		0			0
$817$			10.0540i			0.351744i
$818$	13.8078			0.482777
$819$		0			0
$820$	14.2462			0.497499
$821$			30.8769i			1.07761i	0.842430	+	0.538806i	$0.181124\pi$
$821$			30.8769i			1.07761i	−0.842430	+	0.538806i	$0.818876\pi$
$822$		0			0
$823$	−30.7386			−1.07148			−0.535741	−	0.844383i	$-0.679968\pi$
$823$	−30.7386			−1.07148			−0.535741	+	0.844383i	$0.679968\pi$
$824$			1.80776i			0.0629764i
$825$		0			0
$826$			4.24621i			0.147745i
$827$		−	22.0540i		−	0.766892i	−0.923563	−	0.383446i	$-0.874737\pi$
$827$		−	22.0540i		−	0.766892i	0.923563	−	0.383446i	$-0.125263\pi$
$828$		0			0
$829$	42.0540			1.46059			0.730297	−	0.683129i	$-0.239381\pi$
$829$	42.0540			1.46059			0.730297	+	0.683129i	$0.239381\pi$
$830$		−	7.12311i		−	0.247247i
$831$		0			0
$832$	−3.56155	+	0.561553i	−0.123475	+	0.0194683i
$833$	−6.68466			−0.231610
$834$		0			0
$835$	42.0540			1.45534
$836$	8.68466			0.300365
$837$		0			0
$838$		−	20.6847i		−	0.714540i
$839$		−	25.7538i		−	0.889120i	−0.895749	−	0.444560i	$-0.853360\pi$
$839$		−	25.7538i		−	0.889120i	0.895749	−	0.444560i	$-0.146640\pi$
$840$		0			0
$841$	−26.5616			−0.915916
$842$	−10.0000			−0.344623
$843$		0			0
$844$	−7.80776			−0.268754
$845$	−14.2462	−	44.0540i	−0.490085	−	1.51550i
$846$		0			0
$847$			19.9309i			0.684833i
$848$	−4.87689			−0.167473
$849$		0			0
$850$			51.3693i			1.76195i
$851$			71.4233i			2.44836i
$852$		0			0
$853$			10.8769i			0.372418i	0.982510	+	0.186209i	$0.0596201\pi$
$853$			10.8769i			0.372418i	−0.982510	+	0.186209i	$0.940380\pi$
$854$	1.56155			0.0534352
$855$		0			0
$856$			4.87689i			0.166689i
$857$	38.0000			1.29806			0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000			1.29806			0.649028	+	0.760765i	$0.275176\pi$
$858$		0			0
$859$	0.384472			0.0131180			0.00655901	−	0.999978i	$-0.497912\pi$
$859$	0.384472			0.0131180			0.00655901	+	0.999978i	$0.497912\pi$
$860$		−	22.9309i		−	0.781936i
$861$		0			0
$862$	−29.8617			−1.01709
$863$			42.7386i			1.45484i	0.686192	+	0.727420i	$0.259281\pi$
$863$			42.7386i			1.45484i	−0.686192	+	0.727420i	$0.740719\pi$
$864$		0			0
$865$		−	13.3693i		−	0.454570i
$866$		−	23.3693i		−	0.794122i
$867$		0			0
$868$	6.24621			0.212010
$869$		−	88.9848i		−	3.01860i
$870$		0			0
$871$	−0.630683	−	4.00000i	−0.0213699	−	0.135535i
$872$	−12.9309			−0.437895
$873$		0			0
$874$	−10.4384			−0.353086
$875$	9.56155			0.323239
$876$		0			0
$877$		−	22.9848i		−	0.776143i	−0.921629	−	0.388072i	$-0.873141\pi$
$877$		−	22.9848i		−	0.776143i	0.921629	−	0.388072i	$-0.126859\pi$
$878$			8.93087i			0.301402i
$879$		0			0
$880$	−19.8078			−0.667720
$881$	−14.3002			−0.481786			−0.240893	−	0.970552i	$-0.577440\pi$
$881$	−14.3002			−0.481786			−0.240893	+	0.970552i	$0.577440\pi$
$882$		0			0
$883$	−14.4384			−0.485892			−0.242946	−	0.970040i	$-0.578114\pi$
$883$	−14.4384			−0.485892			−0.242946	+	0.970040i	$0.578114\pi$
$884$	−23.8078	+	3.75379i	−0.800742	+	0.126254i
$885$		0			0
$886$			6.63068i			0.222762i
$887$	37.8617			1.27127			0.635636	−	0.771989i	$-0.280738\pi$
$887$	37.8617			1.27127			0.635636	+	0.771989i	$0.280738\pi$
$888$		0			0
$889$		−	10.2462i		−	0.343647i
$890$		−	28.4924i		−	0.955068i
$891$		0			0
$892$			1.75379i			0.0587212i
$893$	16.0000			0.535420
$894$		0			0
$895$			39.6155i			1.32420i
$896$	1.00000			0.0334077
$897$		0			0
$898$	9.31534			0.310857
$899$			9.75379i			0.325307i
$900$		0			0
$901$	−32.6004			−1.08608
$902$			22.2462i			0.740718i
$903$		0			0
$904$		−	20.2462i		−	0.673379i
$905$			47.6155i			1.58279i
$906$		0			0
$907$	−24.4924			−0.813258			−0.406629	−	0.913593i	$-0.633296\pi$
$907$	−24.4924			−0.813258			−0.406629	+	0.913593i	$0.633296\pi$
$908$		−	17.6155i		−	0.584592i
$909$		0			0
$910$	2.00000	+	12.6847i	0.0662994	+	0.420492i
$911$	4.43845			0.147052			0.0735262	−	0.997293i	$-0.476575\pi$
$911$	4.43845			0.147052			0.0735262	+	0.997293i	$0.476575\pi$
$912$		0			0
$913$	11.1231			0.368121
$914$	−16.0000			−0.529233
$915$		0			0
$916$		−	2.00000i		−	0.0660819i
$917$		−	1.56155i		−	0.0515670i
$918$		0			0
$919$	−17.8617			−0.589204			−0.294602	−	0.955620i	$-0.595187\pi$
$919$	−17.8617			−0.589204			−0.294602	+	0.955620i	$0.595187\pi$
$920$	23.8078			0.784919
$921$		0			0
$922$	−36.9309			−1.21625
$923$	−5.26137	−	33.3693i	−0.173180	−	1.09836i
$924$		0			0
$925$		−	82.1080i		−	2.69969i
$926$	−30.5464			−1.00382
$927$		0			0
$928$			1.56155i			0.0512605i
$929$			40.8769i			1.34113i	0.741852	+	0.670564i	$0.233948\pi$
$929$			40.8769i			1.34113i	−0.741852	+	0.670564i	$0.766052\pi$
$930$		0			0
$931$			1.56155i			0.0511778i
$932$	2.00000			0.0655122
$933$		0			0
$934$			1.56155i			0.0510956i
$935$	−132.408			−4.33021
$936$		0			0
$937$	−19.8617			−0.648855			−0.324427	−	0.945911i	$-0.605172\pi$
$937$	−19.8617			−0.648855			−0.324427	+	0.945911i	$0.605172\pi$
$938$			1.12311i			0.0366707i
$939$		0			0
$940$	−36.4924			−1.19025
$941$			26.4924i			0.863628i	0.901963	+	0.431814i	$0.142126\pi$
$941$			26.4924i			0.863628i	−0.901963	+	0.431814i	$0.857874\pi$
$942$		0			0
$943$		−	26.7386i		−	0.870730i
$944$		−	4.24621i		−	0.138202i
$945$		0			0
$946$	35.8078			1.16421
$947$		−	27.8078i		−	0.903631i	−0.892111	−	0.451815i	$-0.850777\pi$
$947$		−	27.8078i		−	0.903631i	0.892111	−	0.451815i	$-0.149223\pi$
$948$		0			0
$949$	6.49242	+	41.1771i	0.210753	+	1.33666i
$950$	12.0000			0.389331
$951$		0			0
$952$	6.68466			0.216651
$953$	−0.738634			−0.0239267			−0.0119633	−	0.999928i	$-0.503808\pi$
$953$	−0.738634			−0.0239267			−0.0119633	+	0.999928i	$0.503808\pi$
$954$		0			0
$955$		−	85.6695i		−	2.77220i
$956$			14.2462i			0.460755i
$957$		0			0
$958$	−12.3002			−0.397401
$959$	−6.68466			−0.215859
$960$		0			0
$961$	−8.01515			−0.258553
$962$	38.0540	−	6.00000i	1.22691	−	0.193448i
$963$		0			0
$964$		−	6.00000i		−	0.193247i
$965$	−42.7386			−1.37581
$966$		0			0
$967$			2.93087i			0.0942504i	0.998889	+	0.0471252i	$0.0150060\pi$
$967$			2.93087i			0.0942504i	−0.998889	+	0.0471252i	$0.984994\pi$
$968$		−	19.9309i		−	0.640602i
$969$		0			0
$970$		−	35.6155i		−	1.14355i
$971$	−4.00000			−0.128366			−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000			−0.128366			−0.0641831	+	0.997938i	$0.520444\pi$
$972$		0			0
$973$		−	10.2462i		−	0.328478i
$974$	−13.7538			−0.440700
$975$		0			0
$976$	−1.56155			−0.0499841
$977$			46.7926i			1.49703i	0.663119	+	0.748514i	$0.269232\pi$
$977$			46.7926i			1.49703i	−0.663119	+	0.748514i	$0.730768\pi$
$978$		0			0
$979$	44.4924			1.42198
$980$		−	3.56155i		−	0.113770i
$981$		0			0
$982$		−	10.2462i		−	0.326970i
$983$			2.43845i			0.0777744i	0.999244	+	0.0388872i	$0.0123813\pi$
$983$			2.43845i			0.0777744i	−0.999244	+	0.0388872i	$0.987619\pi$
$984$		0			0
$985$	21.3693			0.680883
$986$			10.4384i			0.332428i
$987$		0			0
$988$	0.876894	+	5.56155i	0.0278977	+	0.176937i
$989$	−43.0388			−1.36855
$990$		0			0
$991$	−30.2462			−0.960803			−0.480401	−	0.877049i	$-0.659509\pi$
$991$	−30.2462			−0.960803			−0.480401	+	0.877049i	$0.659509\pi$
$992$	−6.24621			−0.198317
$993$		0			0
$994$			9.36932i			0.297177i
$995$		−	17.5616i		−	0.556739i
$996$		0			0
$997$	−34.6307			−1.09676			−0.548382	−	0.836228i	$-0.684756\pi$
$997$	−34.6307			−1.09676			−0.548382	+	0.836228i	$0.684756\pi$
$998$	−1.50758			−0.0477215
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.c.h.883.1		4
3.2	odd	2		546.2.c.e.337.4	yes	4
12.11	even	2		4368.2.h.n.337.4		4
13.12	even	2	inner	1638.2.c.h.883.4		4
21.20	even	2		3822.2.c.h.883.3		4
39.5	even	4		7098.2.a.bv.1.2		2
39.8	even	4		7098.2.a.bg.1.1		2
39.38	odd	2		546.2.c.e.337.1	✓	4
156.155	even	2		4368.2.h.n.337.1		4
273.272	even	2		3822.2.c.h.883.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.e.337.1	✓	4	39.38	odd	2
546.2.c.e.337.4	yes	4	3.2	odd	2
1638.2.c.h.883.1		4	1.1	even	1	trivial
1638.2.c.h.883.4		4	13.12	even	2	inner
3822.2.c.h.883.2		4	273.272	even	2
3822.2.c.h.883.3		4	21.20	even	2
4368.2.h.n.337.1		4	156.155	even	2
4368.2.h.n.337.4		4	12.11	even	2
7098.2.a.bg.1.1		2	39.8	even	4
7098.2.a.bv.1.2		2	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.56155i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.56155i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.56155 q^{10} -5.56155i q^{11} +(3.56155 - 0.561553i) q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.68466 q^{17} -1.56155i q^{19} +3.56155i q^{20} -5.56155 q^{22} +6.68466 q^{23} -7.68466 q^{25} +(-0.561553 - 3.56155i) q^{26} +1.00000i q^{28} -1.56155 q^{29} -6.24621i q^{31} -1.00000i q^{32} -6.68466i q^{34} -3.56155 q^{35} +10.6847i q^{37} -1.56155 q^{38} +3.56155 q^{40} -4.00000i q^{41} -6.43845 q^{43} +5.56155i q^{44} -6.68466i q^{46} +10.2462i q^{47} -1.00000 q^{49} +7.68466i q^{50} +(-3.56155 + 0.561553i) q^{52} -4.87689 q^{53} -19.8078 q^{55} +1.00000 q^{56} +1.56155i q^{58} -4.24621i q^{59} -1.56155 q^{61} -6.24621 q^{62} -1.00000 q^{64} +(-2.00000 - 12.6847i) q^{65} -1.12311i q^{67} -6.68466 q^{68} +3.56155i q^{70} -9.36932i q^{71} +11.5616i q^{73} +10.6847 q^{74} +1.56155i q^{76} -5.56155 q^{77} +16.0000 q^{79} -3.56155i q^{80} -4.00000 q^{82} +2.00000i q^{83} -23.8078i q^{85} +6.43845i q^{86} +5.56155 q^{88} +8.00000i q^{89} +(-0.561553 - 3.56155i) q^{91} -6.68466 q^{92} +10.2462 q^{94} -5.56155 q^{95} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 6 q^{10} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 2 q^{17} - 14 q^{22} + 2 q^{23} - 6 q^{25} + 6 q^{26} + 2 q^{29} - 6 q^{35} + 2 q^{38} + 6 q^{40} - 34 q^{43} - 4 q^{49} - 6 q^{52} - 36 q^{53} - 38 q^{55} + 4 q^{56} + 2 q^{61} + 8 q^{62} - 4 q^{64} - 8 q^{65} - 2 q^{68} + 18 q^{74} - 14 q^{77} + 64 q^{79} - 16 q^{82} + 14 q^{88} + 6 q^{91} - 2 q^{92} + 8 q^{94} - 14 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 6 * q^10 + 6 * q^13 - 4 * q^14 + 4 * q^16 + 2 * q^17 - 14 * q^22 + 2 * q^23 - 6 * q^25 + 6 * q^26 + 2 * q^29 - 6 * q^35 + 2 * q^38 + 6 * q^40 - 34 * q^43 - 4 * q^49 - 6 * q^52 - 36 * q^53 - 38 * q^55 + 4 * q^56 + 2 * q^61 + 8 * q^62 - 4 * q^64 - 8 * q^65 - 2 * q^68 + 18 * q^74 - 14 * q^77 + 64 * q^79 - 16 * q^82 + 14 * q^88 + 6 * q^91 - 2 * q^92 + 8 * q^94 - 14 * q^95

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