LMFDB - Embedded newform 1638.2.c.h.883.2

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			883.2
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1638.883
Dual form			1638.2.c.h.883.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +0.561553i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +0.561553i q^{5} -1.00000i q^{7} +1.00000i q^{8} +0.561553 q^{10} -1.43845i q^{11} +(-0.561553 + 3.56155i) q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.68466 q^{17} +2.56155i q^{19} -0.561553i q^{20} -1.43845 q^{22} -5.68466 q^{23} +4.68466 q^{25} +(3.56155 + 0.561553i) q^{26} +1.00000i q^{28} +2.56155 q^{29} +10.2462i q^{31} -1.00000i q^{32} +5.68466i q^{34} +0.561553 q^{35} -1.68466i q^{37} +2.56155 q^{38} -0.561553 q^{40} -4.00000i q^{41} -10.5616 q^{43} +1.43845i q^{44} +5.68466i q^{46} -6.24621i q^{47} -1.00000 q^{49} -4.68466i q^{50} +(0.561553 - 3.56155i) q^{52} -13.1231 q^{53} +0.807764 q^{55} +1.00000 q^{56} -2.56155i q^{58} +12.2462i q^{59} +2.56155 q^{61} +10.2462 q^{62} -1.00000 q^{64} +(-2.00000 - 0.315342i) q^{65} +7.12311i q^{67} +5.68466 q^{68} -0.561553i q^{70} +15.3693i q^{71} +7.43845i q^{73} -1.68466 q^{74} -2.56155i q^{76} -1.43845 q^{77} +16.0000 q^{79} +0.561553i q^{80} -4.00000 q^{82} +2.00000i q^{83} -3.19224i q^{85} +10.5616i q^{86} +1.43845 q^{88} +8.00000i q^{89} +(3.56155 + 0.561553i) q^{91} +5.68466 q^{92} -6.24621 q^{94} -1.43845 q^{95} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			0.561553i			0.251134i	0.992085	+	0.125567i	$0.0400750\pi$
$5$			0.561553i			0.251134i	−0.992085	+	0.125567i	$0.959925\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	0.561553			0.177579
$11$		−	1.43845i		−	0.433708i	−0.976204	−	0.216854i	$-0.930420\pi$
$11$		−	1.43845i		−	0.433708i	0.976204	−	0.216854i	$-0.0695796\pi$
$12$		0			0
$13$	−0.561553	+	3.56155i	−0.155747	+	0.987797i
$13$	−0.561553	+	3.56155i	−0.155747	+	0.987797i
$14$	−1.00000			−0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$	−5.68466			−1.37873			−0.689366	−	0.724413i	$-0.742111\pi$
$17$	−5.68466			−1.37873			−0.689366	+	0.724413i	$0.742111\pi$
$18$		0			0
$19$			2.56155i			0.587661i	0.955858	+	0.293830i	$0.0949300\pi$
$19$			2.56155i			0.587661i	−0.955858	+	0.293830i	$0.905070\pi$
$20$		−	0.561553i		−	0.125567i
$21$		0			0
$22$	−1.43845			−0.306678
$23$	−5.68466			−1.18533			−0.592667	−	0.805448i	$-0.701925\pi$
$23$	−5.68466			−1.18533			−0.592667	+	0.805448i	$0.701925\pi$
$24$		0			0
$25$	4.68466			0.936932
$26$	3.56155	+	0.561553i	0.698478	+	0.110130i
$27$		0			0
$28$			1.00000i			0.188982i
$29$	2.56155			0.475668			0.237834	−	0.971306i	$-0.423563\pi$
$29$	2.56155			0.475668			0.237834	+	0.971306i	$0.423563\pi$
$30$		0			0
$31$			10.2462i			1.84027i	0.391597	+	0.920137i	$0.371923\pi$
$31$			10.2462i			1.84027i	−0.391597	+	0.920137i	$0.628077\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			5.68466i			0.974911i
$35$	0.561553			0.0949197
$36$		0			0
$37$		−	1.68466i		−	0.276956i	−0.990366	−	0.138478i	$-0.955779\pi$
$37$		−	1.68466i		−	0.276956i	0.990366	−	0.138478i	$-0.0442210\pi$
$38$	2.56155			0.415539
$39$		0			0
$40$	−0.561553			−0.0887893
$41$		−	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
$43$	−10.5616			−1.61062			−0.805311	−	0.592853i	$-0.798002\pi$
$43$	−10.5616			−1.61062			−0.805311	+	0.592853i	$0.798002\pi$
$44$			1.43845i			0.216854i
$45$		0			0
$46$			5.68466i			0.838157i
$47$		−	6.24621i		−	0.911104i	−0.890209	−	0.455552i	$-0.849442\pi$
$47$		−	6.24621i		−	0.911104i	0.890209	−	0.455552i	$-0.150558\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		−	4.68466i		−	0.662511i
$51$		0			0
$52$	0.561553	−	3.56155i	0.0778734	−	0.493899i
$53$	−13.1231			−1.80260			−0.901299	−	0.433198i	$-0.857385\pi$
$53$	−13.1231			−1.80260			−0.901299	+	0.433198i	$0.857385\pi$
$54$		0			0
$55$	0.807764			0.108919
$56$	1.00000			0.133631
$57$		0			0
$58$		−	2.56155i		−	0.336348i
$59$			12.2462i			1.59432i	0.603768	+	0.797160i	$0.293666\pi$
$59$			12.2462i			1.59432i	−0.603768	+	0.797160i	$0.706334\pi$
$60$		0			0
$61$	2.56155			0.327973			0.163987	−	0.986463i	$-0.447565\pi$
$61$	2.56155			0.327973			0.163987	+	0.986463i	$0.447565\pi$
$62$	10.2462			1.30127
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.00000	−	0.315342i	−0.248069	−	0.0391133i
$66$		0			0
$67$			7.12311i			0.870226i	0.900376	+	0.435113i	$0.143292\pi$
$67$			7.12311i			0.870226i	−0.900376	+	0.435113i	$0.856708\pi$
$68$	5.68466			0.689366
$69$		0			0
$70$		−	0.561553i		−	0.0671184i
$71$			15.3693i			1.82400i	0.410188	+	0.912001i	$0.365463\pi$
$71$			15.3693i			1.82400i	−0.410188	+	0.912001i	$0.634537\pi$
$72$		0			0
$73$			7.43845i			0.870604i	0.900284	+	0.435302i	$0.143358\pi$
$73$			7.43845i			0.870604i	−0.900284	+	0.435302i	$0.856642\pi$
$74$	−1.68466			−0.195838
$75$		0			0
$76$		−	2.56155i		−	0.293830i
$77$	−1.43845			−0.163926
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$			0.561553i			0.0627835i
$81$		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
$85$		−	3.19224i		−	0.346247i
$86$			10.5616i			1.13888i
$87$		0			0
$88$	1.43845			0.153339
$89$			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
$91$	3.56155	+	0.561553i	0.373352	+	0.0588667i
$92$	5.68466			0.592667
$93$		0			0
$94$	−6.24621			−0.644247
$95$	−1.43845			−0.147582
$96$		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
$100$	−4.68466			−0.468466
$101$	−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
$103$	−18.8078			−1.85318			−0.926592	−	0.376068i	$-0.877276\pi$
$103$	−18.8078			−1.85318			−0.926592	+	0.376068i	$0.877276\pi$
$104$	−3.56155	−	0.561553i	−0.349239	−	0.0550648i
$105$		0			0
$106$			13.1231i			1.27463i
$107$	13.1231			1.26866			0.634329	−	0.773063i	$-0.281276\pi$
$107$	13.1231			1.26866			0.634329	+	0.773063i	$0.281276\pi$
$108$		0			0
$109$		−	15.9309i		−	1.52590i	−0.646457	−	0.762950i	$-0.723750\pi$
$109$		−	15.9309i		−	1.52590i	0.646457	−	0.762950i	$-0.276250\pi$
$110$		−	0.807764i		−	0.0770173i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	−3.75379			−0.353127			−0.176563	−	0.984289i	$-0.556498\pi$
$113$	−3.75379			−0.353127			−0.176563	+	0.984289i	$0.556498\pi$
$114$		0			0
$115$		−	3.19224i		−	0.297678i
$116$	−2.56155			−0.237834
$117$		0			0
$118$	12.2462			1.12736
$119$			5.68466i			0.521112i
$120$		0			0
$121$	8.93087			0.811897
$122$		−	2.56155i		−	0.231912i
$123$		0			0
$124$		−	10.2462i		−	0.920137i
$125$			5.43845i			0.486430i
$126$		0			0
$127$	−6.24621			−0.554262			−0.277131	−	0.960832i	$-0.589384\pi$
$127$	−6.24621			−0.554262			−0.277131	+	0.960832i	$0.589384\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−0.315342	+	2.00000i	−0.0276573	+	0.175412i
$131$	−2.56155			−0.223804			−0.111902	−	0.993719i	$-0.535694\pi$
$131$	−2.56155			−0.223804			−0.111902	+	0.993719i	$0.535694\pi$
$132$		0			0
$133$	2.56155			0.222115
$134$	7.12311			0.615343
$135$		0			0
$136$		−	5.68466i		−	0.487455i
$137$			5.68466i			0.485673i	0.970067	+	0.242837i	$0.0780779\pi$
$137$			5.68466i			0.485673i	−0.970067	+	0.242837i	$0.921922\pi$
$138$		0			0
$139$	−6.24621			−0.529797			−0.264898	−	0.964276i	$-0.585338\pi$
$139$	−6.24621			−0.529797			−0.264898	+	0.964276i	$0.585338\pi$
$140$	−0.561553			−0.0474599
$141$		0			0
$142$	15.3693			1.28976
$143$	5.12311	+	0.807764i	0.428416	+	0.0675486i
$144$		0			0
$145$			1.43845i			0.119457i
$146$	7.43845			0.615610
$147$		0			0
$148$			1.68466i			0.138478i
$149$		−	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
$151$			4.31534i			0.351178i	0.984464	+	0.175589i	$0.0561829\pi$
$151$			4.31534i			0.351178i	−0.984464	+	0.175589i	$0.943817\pi$
$152$	−2.56155			−0.207769
$153$		0			0
$154$			1.43845i			0.115913i
$155$	−5.75379			−0.462155
$156$		0			0
$157$	8.80776			0.702936			0.351468	−	0.936200i	$-0.385683\pi$
$157$	8.80776			0.702936			0.351468	+	0.936200i	$0.385683\pi$
$158$		−	16.0000i		−	1.27289i
$159$		0			0
$160$	0.561553			0.0443946
$161$			5.68466i			0.448014i
$162$		0			0
$163$			0.876894i			0.0686837i	0.999410	+	0.0343418i	$0.0109335\pi$
$163$			0.876894i			0.0686837i	−0.999410	+	0.0343418i	$0.989067\pi$
$164$			4.00000i			0.312348i
$165$		0			0
$166$	2.00000			0.155230
$167$		−	8.80776i		−	0.681565i	−0.940142	−	0.340783i	$-0.889308\pi$
$167$		−	8.80776i		−	0.681565i	0.940142	−	0.340783i	$-0.110692\pi$
$168$		0			0
$169$	−12.3693	−	4.00000i	−0.951486	−	0.307692i
$170$	−3.19224			−0.244833
$171$		0			0
$172$	10.5616			0.805311
$173$	20.2462			1.53929			0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462			1.53929			0.769645	+	0.638471i	$0.220433\pi$
$174$		0			0
$175$		−	4.68466i		−	0.354127i
$176$		−	1.43845i		−	0.108427i
$177$		0			0
$178$	8.00000			0.599625
$179$	−2.87689			−0.215029			−0.107515	−	0.994204i	$-0.534289\pi$
$179$	−2.87689			−0.215029			−0.107515	+	0.994204i	$0.534289\pi$
$180$		0			0
$181$	11.3693			0.845075			0.422537	−	0.906346i	$-0.361140\pi$
$181$	11.3693			0.845075			0.422537	+	0.906346i	$0.361140\pi$
$182$	0.561553	−	3.56155i	0.0416251	−	0.264000i
$183$		0			0
$184$		−	5.68466i		−	0.419079i
$185$	0.946025			0.0695531
$186$		0			0
$187$			8.17708i			0.597967i
$188$			6.24621i			0.455552i
$189$		0			0
$190$			1.43845i			0.104356i
$191$	−13.0540			−0.944553			−0.472276	−	0.881451i	$-0.656568\pi$
$191$	−13.0540			−0.944553			−0.472276	+	0.881451i	$0.656568\pi$
$192$		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
$199$	−23.9309			−1.69641			−0.848207	−	0.529665i	$-0.822318\pi$
$199$	−23.9309			−1.69641			−0.848207	+	0.529665i	$0.822318\pi$
$200$			4.68466i			0.331255i
$201$		0			0
$202$			6.00000i			0.422159i
$203$		−	2.56155i		−	0.179786i
$204$		0			0
$205$	2.24621			0.156882
$206$			18.8078i			1.31040i
$207$		0			0
$208$	−0.561553	+	3.56155i	−0.0389367	+	0.246949i
$209$	3.68466			0.254873
$210$		0			0
$211$	−12.8078			−0.881723			−0.440861	−	0.897575i	$-0.645327\pi$
$211$	−12.8078			−0.881723			−0.440861	+	0.897575i	$0.645327\pi$
$212$	13.1231			0.901299
$213$		0			0
$214$		−	13.1231i		−	0.897077i
$215$		−	5.93087i		−	0.404482i
$216$		0			0
$217$	10.2462			0.695558
$218$	−15.9309			−1.07897
$219$		0			0
$220$	−0.807764			−0.0544594
$221$	3.19224	−	20.2462i	0.214733	−	1.36191i
$222$		0			0
$223$		−	18.2462i		−	1.22186i	−0.791686	−	0.610928i	$-0.790796\pi$
$223$		−	18.2462i		−	1.22186i	0.791686	−	0.610928i	$-0.209204\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$			3.75379i			0.249698i
$227$		−	23.6155i		−	1.56742i	−0.621128	−	0.783709i	$-0.713325\pi$
$227$		−	23.6155i		−	1.56742i	0.621128	−	0.783709i	$-0.286675\pi$
$228$		0			0
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	−3.19224			−0.210490
$231$		0			0
$232$			2.56155i			0.168174i
$233$	−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
$235$	3.50758			0.228809
$236$		−	12.2462i		−	0.797160i
$237$		0			0
$238$	5.68466			0.368482
$239$			2.24621i			0.145295i	0.997358	+	0.0726477i	$0.0231449\pi$
$239$			2.24621i			0.145295i	−0.997358	+	0.0726477i	$0.976855\pi$
$240$		0			0
$241$			6.00000i			0.386494i	0.981150	+	0.193247i	$0.0619019\pi$
$241$			6.00000i			0.386494i	−0.981150	+	0.193247i	$0.938098\pi$
$242$		−	8.93087i		−	0.574098i
$243$		0			0
$244$	−2.56155			−0.163987
$245$		−	0.561553i		−	0.0358763i
$246$		0			0
$247$	−9.12311	−	1.43845i	−0.580489	−	0.0915262i
$248$	−10.2462			−0.650635
$249$		0			0
$250$	5.43845			0.343958
$251$	15.0540			0.950198			0.475099	−	0.879932i	$-0.342412\pi$
$251$	15.0540			0.950198			0.475099	+	0.879932i	$0.342412\pi$
$252$		0			0
$253$			8.17708i			0.514089i
$254$			6.24621i			0.391922i
$255$		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
$259$	−1.68466			−0.104680
$260$	2.00000	+	0.315342i	0.124035	+	0.0195567i
$261$		0			0
$262$			2.56155i			0.158253i
$263$	−1.36932			−0.0844357			−0.0422178	−	0.999108i	$-0.513442\pi$
$263$	−1.36932			−0.0844357			−0.0422178	+	0.999108i	$0.513442\pi$
$264$		0			0
$265$		−	7.36932i		−	0.452694i
$266$		−	2.56155i		−	0.157059i
$267$		0			0
$268$		−	7.12311i		−	0.435113i
$269$	−6.63068			−0.404280			−0.202140	−	0.979357i	$-0.564790\pi$
$269$	−6.63068			−0.404280			−0.202140	+	0.979357i	$0.564790\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$	−5.68466			−0.344683
$273$		0			0
$274$	5.68466			0.343423
$275$		−	6.73863i		−	0.406355i
$276$		0			0
$277$	−19.1231			−1.14900			−0.574498	−	0.818506i	$-0.694803\pi$
$277$	−19.1231			−1.14900			−0.574498	+	0.818506i	$0.694803\pi$
$278$			6.24621i			0.374623i
$279$		0			0
$280$			0.561553i			0.0335592i
$281$		−	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
$283$	−13.1231			−0.780088			−0.390044	−	0.920796i	$-0.627540\pi$
$283$	−13.1231			−0.780088			−0.390044	+	0.920796i	$0.627540\pi$
$284$		−	15.3693i		−	0.912001i
$285$		0			0
$286$	0.807764	−	5.12311i	0.0477641	−	0.302936i
$287$	−4.00000			−0.236113
$288$		0			0
$289$	15.3153			0.900902
$290$	1.43845			0.0844685
$291$		0			0
$292$		−	7.43845i		−	0.435302i
$293$			24.2462i			1.41648i	0.705972	+	0.708239i	$0.250510\pi$
$293$			24.2462i			1.41648i	−0.705972	+	0.708239i	$0.749490\pi$
$294$		0			0
$295$	−6.87689			−0.400388
$296$	1.68466			0.0979188
$297$		0			0
$298$	−10.0000			−0.579284
$299$	3.19224	−	20.2462i	0.184612	−	1.17087i
$300$		0			0
$301$			10.5616i			0.608758i
$302$	4.31534			0.248320
$303$		0			0
$304$			2.56155i			0.146915i
$305$			1.43845i			0.0823652i
$306$		0			0
$307$			9.75379i			0.556678i	0.960483	+	0.278339i	$0.0897839\pi$
$307$			9.75379i			0.556678i	−0.960483	+	0.278339i	$0.910216\pi$
$308$	1.43845			0.0819631
$309$		0			0
$310$			5.75379i			0.326793i
$311$	−32.4924			−1.84248			−0.921238	−	0.388999i	$-0.872821\pi$
$311$	−32.4924			−1.84248			−0.921238	+	0.388999i	$0.872821\pi$
$312$		0			0
$313$	15.7538			0.890457			0.445228	−	0.895417i	$-0.353122\pi$
$313$	15.7538			0.890457			0.445228	+	0.895417i	$0.353122\pi$
$314$		−	8.80776i		−	0.497051i
$315$		0			0
$316$	−16.0000			−0.900070
$317$		−	21.3693i		−	1.20022i	−0.799917	−	0.600110i	$-0.795123\pi$
$317$		−	21.3693i		−	1.20022i	0.799917	−	0.600110i	$-0.204877\pi$
$318$		0			0
$319$		−	3.68466i		−	0.206301i
$320$		−	0.561553i		−	0.0313918i
$321$		0			0
$322$	5.68466			0.316794
$323$		−	14.5616i		−	0.810226i
$324$		0			0
$325$	−2.63068	+	16.6847i	−0.145924	+	0.925498i
$326$	0.876894			0.0485667
$327$		0			0
$328$	4.00000			0.220863
$329$	−6.24621			−0.344365
$330$		0			0
$331$			7.12311i			0.391521i	0.980652	+	0.195761i	$0.0627176\pi$
$331$			7.12311i			0.391521i	−0.980652	+	0.195761i	$0.937282\pi$
$332$		−	2.00000i		−	0.109764i
$333$		0			0
$334$	−8.80776			−0.481939
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−17.0540			−0.928989			−0.464495	−	0.885576i	$-0.653764\pi$
$337$	−17.0540			−0.928989			−0.464495	+	0.885576i	$0.653764\pi$
$338$	−4.00000	+	12.3693i	−0.217571	+	0.672802i
$339$		0			0
$340$			3.19224i			0.173123i
$341$	14.7386			0.798142
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		−	10.5616i		−	0.569441i
$345$		0			0
$346$		−	20.2462i		−	1.08844i
$347$	−8.49242			−0.455897			−0.227949	−	0.973673i	$-0.573202\pi$
$347$	−8.49242			−0.455897			−0.227949	+	0.973673i	$0.573202\pi$
$348$		0			0
$349$		−	21.3693i		−	1.14387i	−0.820298	−	0.571937i	$-0.806192\pi$
$349$		−	21.3693i		−	1.14387i	0.820298	−	0.571937i	$-0.193808\pi$
$350$	−4.68466			−0.250406
$351$		0			0
$352$	−1.43845			−0.0766695
$353$			1.75379i			0.0933448i	0.998910	+	0.0466724i	$0.0148617\pi$
$353$			1.75379i			0.0933448i	−0.998910	+	0.0466724i	$0.985138\pi$
$354$		0			0
$355$	−8.63068			−0.458069
$356$		−	8.00000i		−	0.423999i
$357$		0			0
$358$			2.87689i			0.152049i
$359$			17.6155i			0.929712i	0.885386	+	0.464856i	$0.153894\pi$
$359$			17.6155i			0.929712i	−0.885386	+	0.464856i	$0.846106\pi$
$360$		0			0
$361$	12.4384			0.654655
$362$		−	11.3693i		−	0.597558i
$363$		0			0
$364$	−3.56155	−	0.561553i	−0.186676	−	0.0294334i
$365$	−4.17708			−0.218638
$366$		0			0
$367$	25.3693			1.32427			0.662134	−	0.749386i	$-0.269651\pi$
$367$	25.3693			1.32427			0.662134	+	0.749386i	$0.269651\pi$
$368$	−5.68466			−0.296333
$369$		0			0
$370$		−	0.946025i		−	0.0491815i
$371$			13.1231i			0.681318i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$	8.17708			0.422827
$375$		0			0
$376$	6.24621			0.322124
$377$	−1.43845	+	9.12311i	−0.0740838	+	0.469864i
$378$		0			0
$379$		−	27.1231i		−	1.39322i	−0.717450	−	0.696610i	$-0.754691\pi$
$379$		−	27.1231i		−	1.39322i	0.717450	−	0.696610i	$-0.245309\pi$
$380$	1.43845			0.0737908
$381$		0			0
$382$			13.0540i			0.667899i
$383$		−	13.4384i		−	0.686673i	−0.939213	−	0.343336i	$-0.888443\pi$
$383$		−	13.4384i		−	0.686673i	0.939213	−	0.343336i	$-0.111557\pi$
$384$		0			0
$385$		−	0.807764i		−	0.0411675i
$386$	−12.0000			−0.610784
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	−18.8769			−0.957097			−0.478548	−	0.878061i	$-0.658837\pi$
$389$	−18.8769			−0.957097			−0.478548	+	0.878061i	$0.658837\pi$
$390$		0			0
$391$	32.3153			1.63426
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	6.00000			0.302276
$395$			8.98485i			0.452077i
$396$		0			0
$397$		−	9.36932i		−	0.470233i	−0.971967	−	0.235116i	$-0.924453\pi$
$397$		−	9.36932i		−	0.470233i	0.971967	−	0.235116i	$-0.0755471\pi$
$398$			23.9309i			1.19955i
$399$		0			0
$400$	4.68466			0.234233
$401$		−	34.9848i		−	1.74706i	−0.486771	−	0.873530i	$-0.661825\pi$
$401$		−	34.9848i		−	1.74706i	0.486771	−	0.873530i	$-0.338175\pi$
$402$		0			0
$403$	−36.4924	−	5.75379i	−1.81782	−	0.286617i
$404$	6.00000			0.298511
$405$		0			0
$406$	−2.56155			−0.127128
$407$	−2.42329			−0.120118
$408$		0			0
$409$		−	6.80776i		−	0.336622i	−0.985734	−	0.168311i	$-0.946169\pi$
$409$		−	6.80776i		−	0.336622i	0.985734	−	0.168311i	$-0.0538313\pi$
$410$		−	2.24621i		−	0.110932i
$411$		0			0
$412$	18.8078			0.926592
$413$	12.2462			0.602597
$414$		0			0
$415$	−1.12311			−0.0551311
$416$	3.56155	+	0.561553i	0.174619	+	0.0275324i
$417$		0			0
$418$		−	3.68466i		−	0.180223i
$419$	8.31534			0.406231			0.203116	−	0.979155i	$-0.434893\pi$
$419$	8.31534			0.406231			0.203116	+	0.979155i	$0.434893\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$			12.8078i			0.623472i
$423$		0			0
$424$		−	13.1231i		−	0.637314i
$425$	−26.6307			−1.29178
$426$		0			0
$427$		−	2.56155i		−	0.123962i
$428$	−13.1231			−0.634329
$429$		0			0
$430$	−5.93087			−0.286012
$431$			27.8617i			1.34205i	0.741433	+	0.671026i	$0.234146\pi$
$431$			27.8617i			1.34205i	−0.741433	+	0.671026i	$0.765854\pi$
$432$		0			0
$433$	−1.36932			−0.0658052			−0.0329026	−	0.999459i	$-0.510475\pi$
$433$	−1.36932			−0.0658052			−0.0329026	+	0.999459i	$0.510475\pi$
$434$		−	10.2462i		−	0.491834i
$435$		0			0
$436$			15.9309i			0.762950i
$437$		−	14.5616i		−	0.696574i
$438$		0			0
$439$	19.9309			0.951249			0.475624	−	0.879649i	$-0.342222\pi$
$439$	19.9309			0.951249			0.475624	+	0.879649i	$0.342222\pi$
$440$			0.807764i			0.0385086i
$441$		0			0
$442$	−20.2462	−	3.19224i	−0.963014	−	0.151839i
$443$	−31.3693			−1.49040			−0.745201	−	0.666840i	$-0.767646\pi$
$443$	−31.3693			−1.49040			−0.745201	+	0.666840i	$0.767646\pi$
$444$		0			0
$445$	−4.49242			−0.212961
$446$	−18.2462			−0.863983
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			21.6847i			1.02336i	0.859175	+	0.511681i	$0.170977\pi$
$449$			21.6847i			1.02336i	−0.859175	+	0.511681i	$0.829023\pi$
$450$		0			0
$451$	−5.75379			−0.270935
$452$	3.75379			0.176563
$453$		0			0
$454$	−23.6155			−1.10833
$455$	−0.315342	+	2.00000i	−0.0147834	+	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
$460$			3.19224i			0.148839i
$461$		−	8.06913i		−	0.375817i	−0.982187	−	0.187908i	$-0.939829\pi$
$461$		−	8.06913i		−	0.375817i	0.982187	−	0.187908i	$-0.0601708\pi$
$462$		0			0
$463$			39.5464i			1.83788i	0.394401	+	0.918938i	$0.370952\pi$
$463$			39.5464i			1.83788i	−0.394401	+	0.918938i	$0.629048\pi$
$464$	2.56155			0.118917
$465$		0			0
$466$			2.00000i			0.0926482i
$467$	2.56155			0.118535			0.0592673	−	0.998242i	$-0.481124\pi$
$467$	2.56155			0.118535			0.0592673	+	0.998242i	$0.481124\pi$
$468$		0			0
$469$	7.12311			0.328914
$470$		−	3.50758i		−	0.161792i
$471$		0			0
$472$	−12.2462			−0.563678
$473$			15.1922i			0.698540i
$474$		0			0
$475$			12.0000i			0.550598i
$476$		−	5.68466i		−	0.260556i
$477$		0			0
$478$	2.24621			0.102739
$479$			41.3002i			1.88705i	0.331296	+	0.943527i	$0.392514\pi$
$479$			41.3002i			1.88705i	−0.331296	+	0.943527i	$0.607486\pi$
$480$		0			0
$481$	6.00000	+	0.946025i	0.273576	+	0.0431350i
$482$	6.00000			0.273293
$483$		0			0
$484$	−8.93087			−0.405949
$485$	−5.61553			−0.254988
$486$		0			0
$487$		−	30.2462i		−	1.37059i	−0.728267	−	0.685293i	$-0.759674\pi$
$487$		−	30.2462i		−	1.37059i	0.728267	−	0.685293i	$-0.240326\pi$
$488$			2.56155i			0.115956i
$489$		0			0
$490$	−0.561553			−0.0253684
$491$	−6.24621			−0.281888			−0.140944	−	0.990018i	$-0.545014\pi$
$491$	−6.24621			−0.281888			−0.140944	+	0.990018i	$0.545014\pi$
$492$		0			0
$493$	−14.5616			−0.655819
$494$	−1.43845	+	9.12311i	−0.0647188	+	0.410468i
$495$		0			0
$496$			10.2462i			0.460068i
$497$	15.3693			0.689408
$498$		0			0
$499$		−	34.4924i		−	1.54409i	−0.635566	−	0.772046i	$-0.719233\pi$
$499$		−	34.4924i		−	1.54409i	0.635566	−	0.772046i	$-0.280767\pi$
$500$		−	5.43845i		−	0.243215i
$501$		0			0
$502$		−	15.0540i		−	0.671892i
$503$	5.61553			0.250384			0.125192	−	0.992133i	$-0.460045\pi$
$503$	5.61553			0.250384			0.125192	+	0.992133i	$0.460045\pi$
$504$		0			0
$505$		−	3.36932i		−	0.149933i
$506$	8.17708			0.363516
$507$		0			0
$508$	6.24621			0.277131
$509$			9.68466i			0.429265i	0.976695	+	0.214632i	$0.0688554\pi$
$509$			9.68466i			0.429265i	−0.976695	+	0.214632i	$0.931145\pi$
$510$		0			0
$511$	7.43845			0.329058
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	26.0000i		−	1.14681i
$515$		−	10.5616i		−	0.465398i
$516$		0			0
$517$	−8.98485			−0.395153
$518$			1.68466i			0.0740196i
$519$		0			0
$520$	0.315342	−	2.00000i	0.0138286	−	0.0877058i
$521$	29.0540			1.27288			0.636439	−	0.771327i	$-0.280407\pi$
$521$	29.0540			1.27288			0.636439	+	0.771327i	$0.280407\pi$
$522$		0			0
$523$	24.4924			1.07098			0.535489	−	0.844542i	$-0.320127\pi$
$523$	24.4924			1.07098			0.535489	+	0.844542i	$0.320127\pi$
$524$	2.56155			0.111902
$525$		0			0
$526$			1.36932i			0.0597051i
$527$		−	58.2462i		−	2.53724i
$528$		0			0
$529$	9.31534			0.405015
$530$	−7.36932			−0.320103
$531$		0			0
$532$	−2.56155			−0.111057
$533$	14.2462	+	2.24621i	0.617072	+	0.0972942i
$534$		0			0
$535$			7.36932i			0.318603i
$536$	−7.12311			−0.307671
$537$		0			0
$538$			6.63068i			0.285869i
$539$			1.43845i			0.0619583i
$540$		0			0
$541$		−	26.8078i		−	1.15256i	−0.817254	−	0.576278i	$-0.804505\pi$
$541$		−	26.8078i		−	1.15256i	0.817254	−	0.576278i	$-0.195495\pi$
$542$	−8.00000			−0.343629
$543$		0			0
$544$			5.68466i			0.243728i
$545$	8.94602			0.383206
$546$		0			0
$547$	−36.9848			−1.58136			−0.790679	−	0.612231i	$-0.790272\pi$
$547$	−36.9848			−1.58136			−0.790679	+	0.612231i	$0.790272\pi$
$548$		−	5.68466i		−	0.242837i
$549$		0			0
$550$	−6.73863			−0.287336
$551$			6.56155i			0.279532i
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$			19.1231i			0.812463i
$555$		0			0
$556$	6.24621			0.264898
$557$			0.246211i			0.0104323i	0.999986	+	0.00521615i	$0.00166036\pi$
$557$			0.246211i			0.0104323i	−0.999986	+	0.00521615i	$0.998340\pi$
$558$		0			0
$559$	5.93087	−	37.6155i	0.250849	−	1.59097i
$560$	0.561553			0.0237299
$561$		0			0
$562$	−6.00000			−0.253095
$563$	−0.946025			−0.0398702			−0.0199351	−	0.999801i	$-0.506346\pi$
$563$	−0.946025			−0.0398702			−0.0199351	+	0.999801i	$0.506346\pi$
$564$		0			0
$565$		−	2.10795i		−	0.0886821i
$566$			13.1231i			0.551605i
$567$		0			0
$568$	−15.3693			−0.644882
$569$	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
$571$	−1.75379			−0.0733938			−0.0366969	−	0.999326i	$-0.511684\pi$
$571$	−1.75379			−0.0733938			−0.0366969	+	0.999326i	$0.511684\pi$
$572$	−5.12311	−	0.807764i	−0.214208	−	0.0337743i
$573$		0			0
$574$			4.00000i			0.166957i
$575$	−26.6307			−1.11058
$576$		0			0
$577$			24.2462i			1.00938i	0.863300	+	0.504691i	$0.168394\pi$
$577$			24.2462i			1.00938i	−0.863300	+	0.504691i	$0.831606\pi$
$578$		−	15.3153i		−	0.637034i
$579$		0			0
$580$		−	1.43845i		−	0.0597283i
$581$	2.00000			0.0829740
$582$		0			0
$583$			18.8769i			0.781801i
$584$	−7.43845			−0.307805
$585$		0			0
$586$	24.2462			1.00160
$587$			28.8769i			1.19188i	0.803030	+	0.595938i	$0.203220\pi$
$587$			28.8769i			1.19188i	−0.803030	+	0.595938i	$0.796780\pi$
$588$		0			0
$589$	−26.2462			−1.08146
$590$			6.87689i			0.283117i
$591$		0			0
$592$		−	1.68466i		−	0.0692390i
$593$			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
$595$	−3.19224			−0.130869
$596$			10.0000i			0.409616i
$597$		0			0
$598$	−20.2462	−	3.19224i	−0.827929	−	0.130540i
$599$	23.3002			0.952020			0.476010	−	0.879440i	$-0.342083\pi$
$599$	23.3002			0.952020			0.476010	+	0.879440i	$0.342083\pi$
$600$		0			0
$601$	25.8617			1.05492			0.527461	−	0.849579i	$-0.323144\pi$
$601$	25.8617			1.05492			0.527461	+	0.849579i	$0.323144\pi$
$602$	10.5616			0.430457
$603$		0			0
$604$		−	4.31534i		−	0.175589i
$605$			5.01515i			0.203895i
$606$		0			0
$607$	41.5464			1.68632			0.843158	−	0.537666i	$-0.180694\pi$
$607$	41.5464			1.68632			0.843158	+	0.537666i	$0.180694\pi$
$608$	2.56155			0.103885
$609$		0			0
$610$	1.43845			0.0582410
$611$	22.2462	+	3.50758i	0.899985	+	0.141901i
$612$		0			0
$613$			18.8078i			0.759638i	0.925061	+	0.379819i	$0.124014\pi$
$613$			18.8078i			0.759638i	−0.925061	+	0.379819i	$0.875986\pi$
$614$	9.75379			0.393631
$615$		0			0
$616$		−	1.43845i		−	0.0579567i
$617$		−	26.8078i		−	1.07924i	−0.841909	−	0.539620i	$-0.818568\pi$
$617$		−	26.8078i		−	1.07924i	0.841909	−	0.539620i	$-0.181432\pi$
$618$		0			0
$619$		−	2.06913i		−	0.0831654i	−0.999135	−	0.0415827i	$-0.986760\pi$
$619$		−	2.06913i		−	0.0831654i	0.999135	−	0.0415827i	$-0.0132400\pi$
$620$	5.75379			0.231078
$621$		0			0
$622$			32.4924i			1.30283i
$623$	8.00000			0.320513
$624$		0			0
$625$	20.3693			0.814773
$626$		−	15.7538i		−	0.629648i
$627$		0			0
$628$	−8.80776			−0.351468
$629$			9.57671i			0.381848i
$630$		0			0
$631$			0.315342i			0.0125535i	0.999980	+	0.00627677i	$0.00199797\pi$
$631$			0.315342i			0.0125535i	−0.999980	+	0.00627677i	$0.998002\pi$
$632$			16.0000i			0.636446i
$633$		0			0
$634$	−21.3693			−0.848684
$635$		−	3.50758i		−	0.139194i
$636$		0			0
$637$	0.561553	−	3.56155i	0.0222495	−	0.141114i
$638$	−3.68466			−0.145877
$639$		0			0
$640$	−0.561553			−0.0221973
$641$	44.1080			1.74216			0.871080	−	0.491142i	$-0.163420\pi$
$641$	44.1080			1.74216			0.871080	+	0.491142i	$0.163420\pi$
$642$		0			0
$643$		−	20.8078i		−	0.820578i	−0.911956	−	0.410289i	$-0.865428\pi$
$643$		−	20.8078i		−	0.820578i	0.911956	−	0.410289i	$-0.134572\pi$
$644$		−	5.68466i		−	0.224007i
$645$		0			0
$646$	−14.5616			−0.572917
$647$	−35.3693			−1.39051			−0.695256	−	0.718763i	$-0.744709\pi$
$647$	−35.3693			−1.39051			−0.695256	+	0.718763i	$0.744709\pi$
$648$		0			0
$649$	17.6155			0.691470
$650$	16.6847	+	2.63068i	0.654426	+	0.103184i
$651$		0			0
$652$		−	0.876894i		−	0.0343418i
$653$	25.4384			0.995483			0.497742	−	0.867325i	$-0.334163\pi$
$653$	25.4384			0.995483			0.497742	+	0.867325i	$0.334163\pi$
$654$		0			0
$655$		−	1.43845i		−	0.0562048i
$656$		−	4.00000i		−	0.156174i
$657$		0			0
$658$			6.24621i			0.243503i
$659$	−21.1231			−0.822839			−0.411420	−	0.911446i	$-0.634967\pi$
$659$	−21.1231			−0.822839			−0.411420	+	0.911446i	$0.634967\pi$
$660$		0			0
$661$			4.73863i			0.184311i	0.995745	+	0.0921557i	$0.0293758\pi$
$661$			4.73863i			0.184311i	−0.995745	+	0.0921557i	$0.970624\pi$
$662$	7.12311			0.276847
$663$		0			0
$664$	−2.00000			−0.0776151
$665$			1.43845i			0.0557806i
$666$		0			0
$667$	−14.5616			−0.563826
$668$			8.80776i			0.340783i
$669$		0			0
$670$			4.00000i			0.154533i
$671$		−	3.68466i		−	0.142245i
$672$		0			0
$673$	17.1922			0.662712			0.331356	−	0.943506i	$-0.392494\pi$
$673$	17.1922			0.662712			0.331356	+	0.943506i	$0.392494\pi$
$674$			17.0540i			0.656895i
$675$		0			0
$676$	12.3693	+	4.00000i	0.475743	+	0.153846i
$677$	−6.63068			−0.254838			−0.127419	−	0.991849i	$-0.540669\pi$
$677$	−6.63068			−0.254838			−0.127419	+	0.991849i	$0.540669\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$	3.19224			0.122417
$681$		0			0
$682$		−	14.7386i		−	0.564371i
$683$			11.6847i			0.447101i	0.974692	+	0.223551i	$0.0717648\pi$
$683$			11.6847i			0.447101i	−0.974692	+	0.223551i	$0.928235\pi$
$684$		0			0
$685$	−3.19224			−0.121969
$686$	1.00000			0.0381802
$687$		0			0
$688$	−10.5616			−0.402655
$689$	7.36932	−	46.7386i	0.280749	−	1.78060i
$690$		0			0
$691$			3.50758i			0.133435i	0.997772	+	0.0667173i	$0.0212526\pi$
$691$			3.50758i			0.133435i	−0.997772	+	0.0667173i	$0.978747\pi$
$692$	−20.2462			−0.769645
$693$		0			0
$694$			8.49242i			0.322368i
$695$		−	3.50758i		−	0.133050i
$696$		0			0
$697$			22.7386i			0.861287i
$698$	−21.3693			−0.808841
$699$		0			0
$700$			4.68466i			0.177063i
$701$	−37.6155			−1.42072			−0.710359	−	0.703839i	$-0.751468\pi$
$701$	−37.6155			−1.42072			−0.710359	+	0.703839i	$0.751468\pi$
$702$		0			0
$703$	4.31534			0.162756
$704$			1.43845i			0.0542135i
$705$		0			0
$706$	1.75379			0.0660047
$707$			6.00000i			0.225653i
$708$		0			0
$709$			48.2462i			1.81192i	0.423358	+	0.905962i	$0.360851\pi$
$709$			48.2462i			1.81192i	−0.423358	+	0.905962i	$0.639149\pi$
$710$			8.63068i			0.323904i
$711$		0			0
$712$	−8.00000			−0.299813
$713$		−	58.2462i		−	2.18134i
$714$		0			0
$715$	−0.453602	+	2.87689i	−0.0169638	+	0.107590i
$716$	2.87689			0.107515
$717$		0			0
$718$	17.6155			0.657406
$719$	−30.2462			−1.12799			−0.563997	−	0.825777i	$-0.690737\pi$
$719$	−30.2462			−1.12799			−0.563997	+	0.825777i	$0.690737\pi$
$720$		0			0
$721$			18.8078i			0.700438i
$722$		−	12.4384i		−	0.462911i
$723$		0			0
$724$	−11.3693			−0.422537
$725$	12.0000			0.445669
$726$		0			0
$727$	−20.0691			−0.744323			−0.372161	−	0.928168i	$-0.621383\pi$
$727$	−20.0691			−0.744323			−0.372161	+	0.928168i	$0.621383\pi$
$728$	−0.561553	+	3.56155i	−0.0208125	+	0.132000i
$729$		0			0
$730$			4.17708i			0.154601i
$731$	60.0388			2.22062
$732$		0			0
$733$		−	31.1231i		−	1.14956i	−0.818309	−	0.574779i	$-0.805088\pi$
$733$		−	31.1231i		−	1.14956i	0.818309	−	0.574779i	$-0.194912\pi$
$734$		−	25.3693i		−	0.936399i
$735$		0			0
$736$			5.68466i			0.209539i
$737$	10.2462			0.377424
$738$		0			0
$739$			23.6155i			0.868711i	0.900741	+	0.434356i	$0.143024\pi$
$739$			23.6155i			0.868711i	−0.900741	+	0.434356i	$0.856976\pi$
$740$	−0.946025			−0.0347766
$741$		0			0
$742$	13.1231			0.481764
$743$			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
$745$	5.61553			0.205737
$746$			6.00000i			0.219676i
$747$		0			0
$748$		−	8.17708i		−	0.298984i
$749$		−	13.1231i		−	0.479508i
$750$		0			0
$751$	−14.2462			−0.519852			−0.259926	−	0.965629i	$-0.583698\pi$
$751$	−14.2462			−0.519852			−0.259926	+	0.965629i	$0.583698\pi$
$752$		−	6.24621i		−	0.227776i
$753$		0			0
$754$	9.12311	+	1.43845i	0.332244	+	0.0523852i
$755$	−2.42329			−0.0881926
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$	−27.1231			−0.985156
$759$		0			0
$760$		−	1.43845i		−	0.0521780i
$761$			30.8769i			1.11929i	0.828734	+	0.559643i	$0.189062\pi$
$761$			30.8769i			1.11929i	−0.828734	+	0.559643i	$0.810938\pi$
$762$		0			0
$763$	−15.9309			−0.576736
$764$	13.0540			0.472276
$765$		0			0
$766$	−13.4384			−0.485551
$767$	−43.6155	−	6.87689i	−1.57487	−	0.248310i
$768$		0			0
$769$		−	12.5616i		−	0.452981i	−0.974013	−	0.226491i	$-0.927275\pi$
$769$		−	12.5616i		−	0.452981i	0.974013	−	0.226491i	$-0.0727253\pi$
$770$	−0.807764			−0.0291098
$771$		0			0
$772$			12.0000i			0.431889i
$773$			15.9309i			0.572994i	0.958081	+	0.286497i	$0.0924908\pi$
$773$			15.9309i			0.572994i	−0.958081	+	0.286497i	$0.907509\pi$
$774$		0			0
$775$			48.0000i			1.72421i
$776$	−10.0000			−0.358979
$777$		0			0
$778$			18.8769i			0.676769i
$779$	10.2462			0.367109
$780$		0			0
$781$	22.1080			0.791085
$782$		−	32.3153i		−	1.15559i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			4.94602i			0.176531i
$786$		0			0
$787$			45.9309i			1.63726i	0.574322	+	0.818629i	$0.305266\pi$
$787$			45.9309i			1.63726i	−0.574322	+	0.818629i	$0.694734\pi$
$788$		−	6.00000i		−	0.213741i
$789$		0			0
$790$	8.98485			0.319666
$791$			3.75379i			0.133469i
$792$		0			0
$793$	−1.43845	+	9.12311i	−0.0510808	+	0.323971i
$794$	−9.36932			−0.332505
$795$		0			0
$796$	23.9309			0.848207
$797$	−20.2462			−0.717158			−0.358579	−	0.933499i	$-0.616739\pi$
$797$	−20.2462			−0.717158			−0.358579	+	0.933499i	$0.616739\pi$
$798$		0			0
$799$			35.5076i			1.25617i
$800$		−	4.68466i		−	0.165628i
$801$		0			0
$802$	−34.9848			−1.23536
$803$	10.6998			0.377588
$804$		0			0
$805$	−3.19224			−0.112512
$806$	−5.75379	+	36.4924i	−0.202669	+	1.28539i
$807$		0			0
$808$		−	6.00000i		−	0.211079i
$809$	30.6307			1.07692			0.538459	−	0.842652i	$-0.319007\pi$
$809$	30.6307			1.07692			0.538459	+	0.842652i	$0.319007\pi$
$810$		0			0
$811$			35.0540i			1.23091i	0.788171	+	0.615456i	$0.211028\pi$
$811$			35.0540i			1.23091i	−0.788171	+	0.615456i	$0.788972\pi$
$812$			2.56155i			0.0898929i
$813$		0			0
$814$			2.42329i			0.0849363i
$815$	−0.492423			−0.0172488
$816$		0			0
$817$		−	27.0540i		−	0.946499i
$818$	−6.80776			−0.238028
$819$		0			0
$820$	−2.24621			−0.0784411
$821$			39.1231i			1.36541i	0.730696	+	0.682703i	$0.239196\pi$
$821$			39.1231i			1.36541i	−0.730696	+	0.682703i	$0.760804\pi$
$822$		0			0
$823$	18.7386			0.653188			0.326594	−	0.945165i	$-0.394099\pi$
$823$	18.7386			0.653188			0.326594	+	0.945165i	$0.394099\pi$
$824$		−	18.8078i		−	0.655200i
$825$		0			0
$826$		−	12.2462i		−	0.426100i
$827$			15.0540i			0.523478i	0.965139	+	0.261739i	$0.0842960\pi$
$827$			15.0540i			0.523478i	−0.965139	+	0.261739i	$0.915704\pi$
$828$		0			0
$829$	4.94602			0.171783			0.0858913	−	0.996305i	$-0.472626\pi$
$829$	4.94602			0.171783			0.0858913	+	0.996305i	$0.472626\pi$
$830$			1.12311i			0.0389836i
$831$		0			0
$832$	0.561553	−	3.56155i	0.0194683	−	0.123475i
$833$	5.68466			0.196962
$834$		0			0
$835$	4.94602			0.171164
$836$	−3.68466			−0.127437
$837$		0			0
$838$		−	8.31534i		−	0.287249i
$839$		−	42.2462i		−	1.45850i	−0.684247	−	0.729251i	$-0.739869\pi$
$839$		−	42.2462i		−	1.45850i	0.684247	−	0.729251i	$-0.260131\pi$
$840$		0			0
$841$	−22.4384			−0.773740
$842$	−10.0000			−0.344623
$843$		0			0
$844$	12.8078			0.440861
$845$	2.24621	−	6.94602i	0.0772720	−	0.238951i
$846$		0			0
$847$		−	8.93087i		−	0.306868i
$848$	−13.1231			−0.450649
$849$		0			0
$850$			26.6307i			0.913425i
$851$			9.57671i			0.328285i
$852$		0			0
$853$			19.1231i			0.654763i	0.944892	+	0.327381i	$0.106166\pi$
$853$			19.1231i			0.654763i	−0.944892	+	0.327381i	$0.893834\pi$
$854$	−2.56155			−0.0876545
$855$		0			0
$856$			13.1231i			0.448539i
$857$	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
$859$	41.6155			1.41990			0.709952	−	0.704250i	$-0.248717\pi$
$859$	41.6155			1.41990			0.709952	+	0.704250i	$0.248717\pi$
$860$			5.93087i			0.202241i
$861$		0			0
$862$	27.8617			0.948975
$863$		−	6.73863i		−	0.229386i	−0.993401	−	0.114693i	$-0.963412\pi$
$863$		−	6.73863i		−	0.229386i	0.993401	−	0.114693i	$-0.0365884\pi$
$864$		0			0
$865$			11.3693i			0.386568i
$866$			1.36932i			0.0465313i
$867$		0			0
$868$	−10.2462			−0.347779
$869$		−	23.0152i		−	0.780736i
$870$		0			0
$871$	−25.3693	−	4.00000i	−0.859607	−	0.135535i
$872$	15.9309			0.539487
$873$		0			0
$874$	−14.5616			−0.492552
$875$	5.43845			0.183853
$876$		0			0
$877$			42.9848i			1.45150i	0.687961	+	0.725748i	$0.258506\pi$
$877$			42.9848i			1.45150i	−0.687961	+	0.725748i	$0.741494\pi$
$878$		−	19.9309i		−	0.672634i
$879$		0			0
$880$	0.807764			0.0272297
$881$	39.3002			1.32406			0.662028	−	0.749479i	$-0.269696\pi$
$881$	39.3002			1.32406			0.662028	+	0.749479i	$0.269696\pi$
$882$		0			0
$883$	−18.5616			−0.624646			−0.312323	−	0.949976i	$-0.601107\pi$
$883$	−18.5616			−0.624646			−0.312323	+	0.949976i	$0.601107\pi$
$884$	−3.19224	+	20.2462i	−0.107367	+	0.680954i
$885$		0			0
$886$			31.3693i			1.05387i
$887$	−19.8617			−0.666892			−0.333446	−	0.942769i	$-0.608211\pi$
$887$	−19.8617			−0.666892			−0.333446	+	0.942769i	$0.608211\pi$
$888$		0			0
$889$			6.24621i			0.209491i
$890$			4.49242i			0.150586i
$891$		0			0
$892$			18.2462i			0.610928i
$893$	16.0000			0.535420
$894$		0			0
$895$		−	1.61553i		−	0.0540011i
$896$	1.00000			0.0334077
$897$		0			0
$898$	21.6847			0.723626
$899$			26.2462i			0.875360i
$900$		0			0
$901$	74.6004			2.48530
$902$			5.75379i			0.191580i
$903$		0			0
$904$		−	3.75379i		−	0.124849i
$905$			6.38447i			0.212227i
$906$		0			0
$907$	8.49242			0.281986			0.140993	−	0.990011i	$-0.454970\pi$
$907$	8.49242			0.281986			0.140993	+	0.990011i	$0.454970\pi$
$908$			23.6155i			0.783709i
$909$		0			0
$910$	2.00000	+	0.315342i	0.0662994	+	0.0104535i
$911$	8.56155			0.283657			0.141828	−	0.989891i	$-0.454702\pi$
$911$	8.56155			0.283657			0.141828	+	0.989891i	$0.454702\pi$
$912$		0			0
$913$	2.87689			0.0952113
$914$	−16.0000			−0.529233
$915$		0			0
$916$		−	2.00000i		−	0.0660819i
$917$			2.56155i			0.0845899i
$918$		0			0
$919$	39.8617			1.31492			0.657459	−	0.753491i	$-0.271631\pi$
$919$	39.8617			1.31492			0.657459	+	0.753491i	$0.271631\pi$
$920$	3.19224			0.105245
$921$		0			0
$922$	−8.06913			−0.265743
$923$	−54.7386	−	8.63068i	−1.80174	−	0.284082i
$924$		0			0
$925$		−	7.89205i		−	0.259489i
$926$	39.5464			1.29958
$927$		0			0
$928$		−	2.56155i		−	0.0840871i
$929$			49.1231i			1.61168i	0.592136	+	0.805838i	$0.298285\pi$
$929$			49.1231i			1.61168i	−0.592136	+	0.805838i	$0.701715\pi$
$930$		0			0
$931$		−	2.56155i		−	0.0839515i
$932$	2.00000			0.0655122
$933$		0			0
$934$		−	2.56155i		−	0.0838166i
$935$	−4.59186			−0.150170
$936$		0			0
$937$	37.8617			1.23689			0.618445	−	0.785828i	$-0.287763\pi$
$937$	37.8617			1.23689			0.618445	+	0.785828i	$0.287763\pi$
$938$		−	7.12311i		−	0.232578i
$939$		0			0
$940$	−3.50758			−0.114405
$941$		−	6.49242i		−	0.211647i	−0.994385	−	0.105823i	$-0.966252\pi$
$941$		−	6.49242i		−	0.211647i	0.994385	−	0.105823i	$-0.0337478\pi$
$942$		0			0
$943$			22.7386i			0.740472i
$944$			12.2462i			0.398580i
$945$		0			0
$946$	15.1922			0.493942
$947$		−	7.19224i		−	0.233716i	−0.993149	−	0.116858i	$-0.962718\pi$
$947$		−	7.19224i		−	0.233716i	0.993149	−	0.116858i	$-0.0372823\pi$
$948$		0			0
$949$	−26.4924	−	4.17708i	−0.859980	−	0.135594i
$950$	12.0000			0.389331
$951$		0			0
$952$	−5.68466			−0.184241
$953$	48.7386			1.57880			0.789400	−	0.613880i	$-0.210392\pi$
$953$	48.7386			1.57880			0.789400	+	0.613880i	$0.210392\pi$
$954$		0			0
$955$		−	7.33050i		−	0.237209i
$956$		−	2.24621i		−	0.0726477i
$957$		0			0
$958$	41.3002			1.33435
$959$	5.68466			0.183567
$960$		0			0
$961$	−73.9848			−2.38661
$962$	0.946025	−	6.00000i	0.0305011	−	0.193448i
$963$		0			0
$964$		−	6.00000i		−	0.193247i
$965$	6.73863			0.216924
$966$		0			0
$967$		−	25.9309i		−	0.833881i	−0.908934	−	0.416940i	$-0.863102\pi$
$967$		−	25.9309i		−	0.833881i	0.908934	−	0.416940i	$-0.136898\pi$
$968$			8.93087i			0.287049i
$969$		0			0
$970$			5.61553i			0.180304i
$971$	−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
$973$			6.24621i			0.200244i
$974$	−30.2462			−0.969151
$975$		0			0
$976$	2.56155			0.0819933
$977$		−	39.7926i		−	1.27308i	−0.771244	−	0.636539i	$-0.780365\pi$
$977$		−	39.7926i		−	1.27308i	0.771244	−	0.636539i	$-0.219635\pi$
$978$		0			0
$979$	11.5076			0.367784
$980$			0.561553i			0.0179381i
$981$		0			0
$982$			6.24621i			0.199325i
$983$			6.56155i			0.209281i	0.994510	+	0.104641i	$0.0333692\pi$
$983$			6.56155i			0.209281i	−0.994510	+	0.104641i	$0.966631\pi$
$984$		0			0
$985$	−3.36932			−0.107355
$986$			14.5616i			0.463734i
$987$		0			0
$988$	9.12311	+	1.43845i	0.290245	+	0.0457631i
$989$	60.0388			1.90912
$990$		0			0
$991$	−13.7538			−0.436903			−0.218452	−	0.975848i	$-0.570101\pi$
$991$	−13.7538			−0.436903			−0.218452	+	0.975848i	$0.570101\pi$
$992$	10.2462			0.325318
$993$		0			0
$994$		−	15.3693i		−	0.487485i
$995$		−	13.4384i		−	0.426027i
$996$		0			0
$997$	−59.3693			−1.88025			−0.940123	−	0.340837i	$-0.889290\pi$
$997$	−59.3693			−1.88025			−0.940123	+	0.340837i	$0.889290\pi$
$998$	−34.4924			−1.09184
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.c.h.883.2		4
3.2	odd	2		546.2.c.e.337.3	yes	4
12.11	even	2		4368.2.h.n.337.2		4
13.12	even	2	inner	1638.2.c.h.883.3		4
21.20	even	2		3822.2.c.h.883.4		4
39.5	even	4		7098.2.a.bv.1.1		2
39.8	even	4		7098.2.a.bg.1.2		2
39.38	odd	2		546.2.c.e.337.2	✓	4
156.155	even	2		4368.2.h.n.337.3		4
273.272	even	2		3822.2.c.h.883.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.e.337.2	✓	4	39.38	odd	2
546.2.c.e.337.3	yes	4	3.2	odd	2
1638.2.c.h.883.2		4	1.1	even	1	trivial
1638.2.c.h.883.3		4	13.12	even	2	inner
3822.2.c.h.883.1		4	273.272	even	2
3822.2.c.h.883.4		4	21.20	even	2
4368.2.h.n.337.2		4	12.11	even	2
4368.2.h.n.337.3		4	156.155	even	2
7098.2.a.bg.1.2		2	39.8	even	4
7098.2.a.bv.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +0.561553i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +0.561553i q^{5} -1.00000i q^{7} +1.00000i q^{8} +0.561553 q^{10} -1.43845i q^{11} +(-0.561553 + 3.56155i) q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.68466 q^{17} +2.56155i q^{19} -0.561553i q^{20} -1.43845 q^{22} -5.68466 q^{23} +4.68466 q^{25} +(3.56155 + 0.561553i) q^{26} +1.00000i q^{28} +2.56155 q^{29} +10.2462i q^{31} -1.00000i q^{32} +5.68466i q^{34} +0.561553 q^{35} -1.68466i q^{37} +2.56155 q^{38} -0.561553 q^{40} -4.00000i q^{41} -10.5616 q^{43} +1.43845i q^{44} +5.68466i q^{46} -6.24621i q^{47} -1.00000 q^{49} -4.68466i q^{50} +(0.561553 - 3.56155i) q^{52} -13.1231 q^{53} +0.807764 q^{55} +1.00000 q^{56} -2.56155i q^{58} +12.2462i q^{59} +2.56155 q^{61} +10.2462 q^{62} -1.00000 q^{64} +(-2.00000 - 0.315342i) q^{65} +7.12311i q^{67} +5.68466 q^{68} -0.561553i q^{70} +15.3693i q^{71} +7.43845i q^{73} -1.68466 q^{74} -2.56155i q^{76} -1.43845 q^{77} +16.0000 q^{79} +0.561553i q^{80} -4.00000 q^{82} +2.00000i q^{83} -3.19224i q^{85} +10.5616i q^{86} +1.43845 q^{88} +8.00000i q^{89} +(3.56155 + 0.561553i) q^{91} +5.68466 q^{92} -6.24621 q^{94} -1.43845 q^{95} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 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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