LMFDB - Embedded newform 1560.2.g.g.961.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(961,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1560.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.g (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:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.2.g.g.961.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} +3.46410i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} +3.46410i q^{7} +1.00000 q^{9} +(1.00000 + 3.46410i) q^{13} -1.00000i q^{15} -7.46410 q^{17} +7.46410i q^{19} +3.46410i q^{21} -9.46410 q^{23} -1.00000 q^{25} +1.00000 q^{27} +3.46410 q^{29} +2.00000i q^{31} +3.46410 q^{35} -8.00000i q^{37} +(1.00000 + 3.46410i) q^{39} +2.00000i q^{41} +6.92820 q^{43} -1.00000i q^{45} -4.00000i q^{47} -5.00000 q^{49} -7.46410 q^{51} +2.00000 q^{53} +7.46410i q^{57} +14.9282i q^{59} -8.92820 q^{61} +3.46410i q^{63} +(3.46410 - 1.00000i) q^{65} -2.00000i q^{67} -9.46410 q^{69} +2.92820i q^{71} +12.3923i q^{73} -1.00000 q^{75} +10.9282 q^{79} +1.00000 q^{81} -6.92820i q^{83} +7.46410i q^{85} +3.46410 q^{87} -0.928203i q^{89} +(-12.0000 + 3.46410i) q^{91} +2.00000i q^{93} +7.46410 q^{95} -6.53590i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 4 q^{13} - 16 q^{17} - 24 q^{23} - 4 q^{25} + 4 q^{27} + 4 q^{39} - 20 q^{49} - 16 q^{51} + 8 q^{53} - 8 q^{61} - 24 q^{69} - 4 q^{75} + 16 q^{79} + 4 q^{81} - 48 q^{91} + 16 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^13 - 16 * q^17 - 24 * q^23 - 4 * q^25 + 4 * q^27 + 4 * q^39 - 20 * q^49 - 16 * q^51 + 8 * q^53 - 8 * q^61 - 24 * q^69 - 4 * q^75 + 16 * q^79 + 4 * q^81 - 48 * q^91 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			3.46410i			1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$			3.46410i			1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	1.00000	+	3.46410i	0.277350	+	0.960769i
$13$	1.00000	+	3.46410i	0.277350	+	0.960769i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	−7.46410			−1.81031			−0.905155	−	0.425081i	$-0.860246\pi$
$17$	−7.46410			−1.81031			−0.905155	+	0.425081i	$0.860246\pi$
$18$		0			0
$19$			7.46410i			1.71238i	0.516659	+	0.856191i	$0.327175\pi$
$19$			7.46410i			1.71238i	−0.516659	+	0.856191i	$0.672825\pi$
$20$		0			0
$21$			3.46410i			0.755929i
$22$		0			0
$23$	−9.46410			−1.97340			−0.986701	−	0.162547i	$-0.948029\pi$
$23$	−9.46410			−1.97340			−0.986701	+	0.162547i	$0.948029\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.46410			0.643268			0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410			0.643268			0.321634	+	0.946864i	$0.395768\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410			0.585540
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	1.00000	+	3.46410i	0.160128	+	0.554700i
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	6.92820			1.05654			0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820			1.05654			0.528271	+	0.849076i	$0.322841\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	−7.46410			−1.04518
$52$		0			0
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			7.46410i			0.988644i
$58$		0			0
$59$			14.9282i			1.94349i	0.236040	+	0.971743i	$0.424150\pi$
$59$			14.9282i			1.94349i	−0.236040	+	0.971743i	$0.575850\pi$
$60$		0			0
$61$	−8.92820			−1.14314			−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820			−1.14314			−0.571570	+	0.820554i	$0.693665\pi$
$62$		0			0
$63$			3.46410i			0.436436i
$64$		0			0
$65$	3.46410	−	1.00000i	0.429669	−	0.124035i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$	−9.46410			−1.13934
$70$		0			0
$71$			2.92820i			0.347514i	0.984789	+	0.173757i	$0.0555907\pi$
$71$			2.92820i			0.347514i	−0.984789	+	0.173757i	$0.944409\pi$
$72$		0			0
$73$			12.3923i			1.45041i	0.688533	+	0.725205i	$0.258255\pi$
$73$			12.3923i			1.45041i	−0.688533	+	0.725205i	$0.741745\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$		0			0
$78$		0			0
$79$	10.9282			1.22952			0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282			1.22952			0.614759	+	0.788715i	$0.289253\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.92820i		−	0.760469i	−0.924890	−	0.380235i	$-0.875843\pi$
$83$		−	6.92820i		−	0.760469i	0.924890	−	0.380235i	$-0.124157\pi$
$84$		0			0
$85$			7.46410i			0.809595i
$86$		0			0
$87$	3.46410			0.371391
$88$		0			0
$89$		−	0.928203i		−	0.0983893i	−0.998789	−	0.0491947i	$-0.984335\pi$
$89$		−	0.928203i		−	0.0983893i	0.998789	−	0.0491947i	$-0.0156655\pi$
$90$		0			0
$91$	−12.0000	+	3.46410i	−1.25794	+	0.363137i
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	7.46410			0.765801
$96$		0			0
$97$		−	6.53590i		−	0.663620i	−0.943346	−	0.331810i	$-0.892341\pi$
$97$		−	6.53590i		−	0.663620i	0.943346	−	0.331810i	$-0.107659\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.39230			−0.636058			−0.318029	−	0.948081i	$-0.603021\pi$
$101$	−6.39230			−0.636058			−0.318029	+	0.948081i	$0.603021\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$	3.46410			0.338062
$106$		0			0
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$		0			0
$109$			9.46410i			0.906497i	0.891384	+	0.453248i	$0.149735\pi$
$109$			9.46410i			0.906497i	−0.891384	+	0.453248i	$0.850265\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$		0			0
$113$	8.53590			0.802990			0.401495	−	0.915861i	$-0.368491\pi$
$113$	8.53590			0.802990			0.401495	+	0.915861i	$0.368491\pi$
$114$		0			0
$115$			9.46410i			0.882532i
$116$		0			0
$117$	1.00000	+	3.46410i	0.0924500	+	0.320256i
$118$		0			0
$119$		−	25.8564i		−	2.37025i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$			2.00000i			0.180334i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	2.92820			0.259836			0.129918	−	0.991525i	$-0.458529\pi$
$127$	2.92820			0.259836			0.129918	+	0.991525i	$0.458529\pi$
$128$		0			0
$129$	6.92820			0.609994
$130$		0			0
$131$	5.46410			0.477401			0.238700	−	0.971093i	$-0.423279\pi$
$131$	5.46410			0.477401			0.238700	+	0.971093i	$0.423279\pi$
$132$		0			0
$133$	−25.8564			−2.24203
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	14.9282			1.26619			0.633097	−	0.774073i	$-0.281783\pi$
$139$	14.9282			1.26619			0.633097	+	0.774073i	$0.281783\pi$
$140$		0			0
$141$		−	4.00000i		−	0.336861i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	3.46410i		−	0.287678i
$146$		0			0
$147$	−5.00000			−0.412393
$148$		0			0
$149$			0.928203i			0.0760414i	0.999277	+	0.0380207i	$0.0121053\pi$
$149$			0.928203i			0.0760414i	−0.999277	+	0.0380207i	$0.987895\pi$
$150$		0			0
$151$			23.8564i			1.94141i	0.240281	+	0.970703i	$0.422760\pi$
$151$			23.8564i			1.94141i	−0.240281	+	0.970703i	$0.577240\pi$
$152$		0			0
$153$	−7.46410			−0.603437
$154$		0			0
$155$	2.00000			0.160644
$156$		0			0
$157$	−7.85641			−0.627009			−0.313505	−	0.949587i	$-0.601503\pi$
$157$	−7.85641			−0.627009			−0.313505	+	0.949587i	$0.601503\pi$
$158$		0			0
$159$	2.00000			0.158610
$160$		0			0
$161$		−	32.7846i		−	2.58379i
$162$		0			0
$163$		−	8.92820i		−	0.699311i	−0.936878	−	0.349655i	$-0.886299\pi$
$163$		−	8.92820i		−	0.699311i	0.936878	−	0.349655i	$-0.113701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.85641i		−	0.762712i	−0.924428	−	0.381356i	$-0.875457\pi$
$167$		−	9.85641i		−	0.762712i	0.924428	−	0.381356i	$-0.124543\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$			7.46410i			0.570794i
$172$		0			0
$173$	−19.8564			−1.50965			−0.754827	−	0.655924i	$-0.772279\pi$
$173$	−19.8564			−1.50965			−0.754827	+	0.655924i	$0.772279\pi$
$174$		0			0
$175$		−	3.46410i		−	0.261861i
$176$		0			0
$177$			14.9282i			1.12207i
$178$		0			0
$179$	19.3205			1.44408			0.722041	−	0.691850i	$-0.243204\pi$
$179$	19.3205			1.44408			0.722041	+	0.691850i	$0.243204\pi$
$180$		0			0
$181$	−15.8564			−1.17860			−0.589299	−	0.807915i	$-0.700596\pi$
$181$	−15.8564			−1.17860			−0.589299	+	0.807915i	$0.700596\pi$
$182$		0			0
$183$	−8.92820			−0.659992
$184$		0			0
$185$	−8.00000			−0.588172
$186$		0			0
$187$		0			0
$188$		0			0
$189$			3.46410i			0.251976i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$		−	10.5359i		−	0.758391i	−0.925317	−	0.379195i	$-0.876201\pi$
$193$		−	10.5359i		−	0.758391i	0.925317	−	0.379195i	$-0.123799\pi$
$194$		0			0
$195$	3.46410	−	1.00000i	0.248069	−	0.0716115i
$196$		0			0
$197$		−	0.928203i		−	0.0661317i	−0.999453	−	0.0330659i	$-0.989473\pi$
$197$		−	0.928203i		−	0.0661317i	0.999453	−	0.0330659i	$-0.0105271\pi$
$198$		0			0
$199$	16.7846			1.18983			0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846			1.18983			0.594915	+	0.803789i	$0.297186\pi$
$200$		0			0
$201$		−	2.00000i		−	0.141069i
$202$		0			0
$203$			12.0000i			0.842235i
$204$		0			0
$205$	2.00000			0.139686
$206$		0			0
$207$	−9.46410			−0.657801
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.92820			−0.476957			−0.238479	−	0.971148i	$-0.576649\pi$
$211$	−6.92820			−0.476957			−0.238479	+	0.971148i	$0.576649\pi$
$212$		0			0
$213$			2.92820i			0.200637i
$214$		0			0
$215$		−	6.92820i		−	0.472500i
$216$		0			0
$217$	−6.92820			−0.470317
$218$		0			0
$219$			12.3923i			0.837394i
$220$		0			0
$221$	−7.46410	−	25.8564i	−0.502090	−	1.73929i
$222$		0			0
$223$		−	14.3923i		−	0.963780i	−0.876232	−	0.481890i	$-0.839950\pi$
$223$		−	14.3923i		−	0.963780i	0.876232	−	0.481890i	$-0.160050\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$			18.9282i			1.25631i	0.778089	+	0.628154i	$0.216189\pi$
$227$			18.9282i			1.25631i	−0.778089	+	0.628154i	$0.783811\pi$
$228$		0			0
$229$		−	17.4641i		−	1.15406i	−0.816723	−	0.577030i	$-0.804212\pi$
$229$		−	17.4641i		−	1.15406i	0.816723	−	0.577030i	$-0.195788\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	26.3923			1.72902			0.864509	−	0.502618i	$-0.167630\pi$
$233$	26.3923			1.72902			0.864509	+	0.502618i	$0.167630\pi$
$234$		0			0
$235$	−4.00000			−0.260931
$236$		0			0
$237$	10.9282			0.709863
$238$		0			0
$239$			5.07180i			0.328067i	0.986455	+	0.164034i	$0.0524506\pi$
$239$			5.07180i			0.328067i	−0.986455	+	0.164034i	$0.947549\pi$
$240$		0			0
$241$		−	17.8564i		−	1.15023i	−0.818072	−	0.575116i	$-0.804957\pi$
$241$		−	17.8564i		−	1.15023i	0.818072	−	0.575116i	$-0.195043\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			5.00000i			0.319438i
$246$		0			0
$247$	−25.8564	+	7.46410i	−1.64520	+	0.474929i
$248$		0			0
$249$		−	6.92820i		−	0.439057i
$250$		0			0
$251$	−14.5359			−0.917498			−0.458749	−	0.888566i	$-0.651702\pi$
$251$	−14.5359			−0.917498			−0.458749	+	0.888566i	$0.651702\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			7.46410i			0.467420i
$256$		0			0
$257$	13.3205			0.830910			0.415455	−	0.909614i	$-0.363622\pi$
$257$	13.3205			0.830910			0.415455	+	0.909614i	$0.363622\pi$
$258$		0			0
$259$	27.7128			1.72199
$260$		0			0
$261$	3.46410			0.214423
$262$		0			0
$263$	−5.46410			−0.336931			−0.168465	−	0.985708i	$-0.553881\pi$
$263$	−5.46410			−0.336931			−0.168465	+	0.985708i	$0.553881\pi$
$264$		0			0
$265$		−	2.00000i		−	0.122859i
$266$		0			0
$267$		−	0.928203i		−	0.0568051i
$268$		0			0
$269$	8.53590			0.520443			0.260221	−	0.965549i	$-0.416204\pi$
$269$	8.53590			0.520443			0.260221	+	0.965549i	$0.416204\pi$
$270$		0			0
$271$			19.8564i			1.20619i	0.797669	+	0.603095i	$0.206066\pi$
$271$			19.8564i			1.20619i	−0.797669	+	0.603095i	$0.793934\pi$
$272$		0			0
$273$	−12.0000	+	3.46410i	−0.726273	+	0.209657i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−29.7128			−1.78527			−0.892635	−	0.450780i	$-0.851146\pi$
$277$	−29.7128			−1.78527			−0.892635	+	0.450780i	$0.851146\pi$
$278$		0			0
$279$			2.00000i			0.119737i
$280$		0			0
$281$		−	4.92820i		−	0.293992i	−0.989137	−	0.146996i	$-0.953040\pi$
$281$		−	4.92820i		−	0.293992i	0.989137	−	0.146996i	$-0.0469604\pi$
$282$		0			0
$283$	22.9282			1.36294			0.681470	−	0.731846i	$-0.261341\pi$
$283$	22.9282			1.36294			0.681470	+	0.731846i	$0.261341\pi$
$284$		0			0
$285$	7.46410			0.442135
$286$		0			0
$287$	−6.92820			−0.408959
$288$		0			0
$289$	38.7128			2.27722
$290$		0			0
$291$		−	6.53590i		−	0.383141i
$292$		0			0
$293$		−	24.9282i		−	1.45632i	−0.685407	−	0.728161i	$-0.740375\pi$
$293$		−	24.9282i		−	1.45632i	0.685407	−	0.728161i	$-0.259625\pi$
$294$		0			0
$295$	14.9282			0.869154
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.46410	−	32.7846i	−0.547323	−	1.89598i
$300$		0			0
$301$			24.0000i			1.38334i
$302$		0			0
$303$	−6.39230			−0.367228
$304$		0			0
$305$			8.92820i			0.511227i
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−23.7128			−1.34463			−0.672315	−	0.740265i	$-0.734700\pi$
$311$	−23.7128			−1.34463			−0.672315	+	0.740265i	$0.734700\pi$
$312$		0			0
$313$	−15.8564			−0.896257			−0.448129	−	0.893969i	$-0.647909\pi$
$313$	−15.8564			−0.896257			−0.448129	+	0.893969i	$0.647909\pi$
$314$		0			0
$315$	3.46410			0.195180
$316$		0			0
$317$		−	11.8564i		−	0.665922i	−0.942941	−	0.332961i	$-0.891952\pi$
$317$		−	11.8564i		−	0.665922i	0.942941	−	0.332961i	$-0.108048\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$		−	55.7128i		−	3.09994i
$324$		0			0
$325$	−1.00000	−	3.46410i	−0.0554700	−	0.192154i
$326$		0			0
$327$			9.46410i			0.523366i
$328$		0			0
$329$	13.8564			0.763928
$330$		0			0
$331$		−	18.3923i		−	1.01093i	−0.862846	−	0.505466i	$-0.831321\pi$
$331$		−	18.3923i		−	1.01093i	0.862846	−	0.505466i	$-0.168679\pi$
$332$		0			0
$333$		−	8.00000i		−	0.438397i
$334$		0			0
$335$	−2.00000			−0.109272
$336$		0			0
$337$	3.07180			0.167331			0.0836657	−	0.996494i	$-0.473337\pi$
$337$	3.07180			0.167331			0.0836657	+	0.996494i	$0.473337\pi$
$338$		0			0
$339$	8.53590			0.463606
$340$		0			0
$341$		0			0
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$			9.46410i			0.509530i
$346$		0			0
$347$	30.9282			1.66031			0.830156	−	0.557530i	$-0.188251\pi$
$347$	30.9282			1.66031			0.830156	+	0.557530i	$0.188251\pi$
$348$		0			0
$349$			10.5359i			0.563974i	0.959418	+	0.281987i	$0.0909935\pi$
$349$			10.5359i			0.563974i	−0.959418	+	0.281987i	$0.909007\pi$
$350$		0			0
$351$	1.00000	+	3.46410i	0.0533761	+	0.184900i
$352$		0			0
$353$			23.8564i			1.26975i	0.772616	+	0.634874i	$0.218948\pi$
$353$			23.8564i			1.26975i	−0.772616	+	0.634874i	$0.781052\pi$
$354$		0			0
$355$	2.92820			0.155413
$356$		0			0
$357$		−	25.8564i		−	1.36847i
$358$		0			0
$359$			17.0718i			0.901015i	0.892773	+	0.450507i	$0.148757\pi$
$359$			17.0718i			0.901015i	−0.892773	+	0.450507i	$0.851243\pi$
$360$		0			0
$361$	−36.7128			−1.93225
$362$		0			0
$363$	11.0000			0.577350
$364$		0			0
$365$	12.3923			0.648643
$366$		0			0
$367$	2.92820			0.152851			0.0764255	−	0.997075i	$-0.475649\pi$
$367$	2.92820			0.152851			0.0764255	+	0.997075i	$0.475649\pi$
$368$		0			0
$369$			2.00000i			0.104116i
$370$		0			0
$371$			6.92820i			0.359694i
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	3.46410	+	12.0000i	0.178410	+	0.618031i
$378$		0			0
$379$		−	5.60770i		−	0.288048i	−0.989574	−	0.144024i	$-0.953996\pi$
$379$		−	5.60770i		−	0.288048i	0.989574	−	0.144024i	$-0.0460042\pi$
$380$		0			0
$381$	2.92820			0.150016
$382$		0			0
$383$		−	17.8564i		−	0.912420i	−0.889872	−	0.456210i	$-0.849207\pi$
$383$		−	17.8564i		−	0.912420i	0.889872	−	0.456210i	$-0.150793\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.92820			0.352180
$388$		0			0
$389$	33.3205			1.68942			0.844708	−	0.535227i	$-0.179774\pi$
$389$	33.3205			1.68942			0.844708	+	0.535227i	$0.179774\pi$
$390$		0			0
$391$	70.6410			3.57247
$392$		0			0
$393$	5.46410			0.275627
$394$		0			0
$395$		−	10.9282i		−	0.549858i
$396$		0			0
$397$			29.8564i			1.49845i	0.662316	+	0.749225i	$0.269574\pi$
$397$			29.8564i			1.49845i	−0.662316	+	0.749225i	$0.730426\pi$
$398$		0			0
$399$	−25.8564			−1.29444
$400$		0			0
$401$			4.14359i			0.206921i	0.994634	+	0.103461i	$0.0329916\pi$
$401$			4.14359i			0.206921i	−0.994634	+	0.103461i	$0.967008\pi$
$402$		0			0
$403$	−6.92820	+	2.00000i	−0.345118	+	0.0996271i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	12.7846i		−	0.632158i	−0.948733	−	0.316079i	$-0.897633\pi$
$409$		−	12.7846i		−	0.632158i	0.948733	−	0.316079i	$-0.102367\pi$
$410$		0			0
$411$			6.00000i			0.295958i
$412$		0			0
$413$	−51.7128			−2.54462
$414$		0			0
$415$	−6.92820			−0.340092
$416$		0			0
$417$	14.9282			0.731037
$418$		0			0
$419$	−32.3923			−1.58247			−0.791234	−	0.611514i	$-0.790561\pi$
$419$	−32.3923			−1.58247			−0.791234	+	0.611514i	$0.790561\pi$
$420$		0			0
$421$		−	1.46410i		−	0.0713559i	−0.999363	−	0.0356780i	$-0.988641\pi$
$421$		−	1.46410i		−	0.0713559i	0.999363	−	0.0356780i	$-0.0113591\pi$
$422$		0			0
$423$		−	4.00000i		−	0.194487i
$424$		0			0
$425$	7.46410			0.362062
$426$		0			0
$427$		−	30.9282i		−	1.49672i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	9.07180i		−	0.436973i	−0.975840	−	0.218487i	$-0.929888\pi$
$431$		−	9.07180i		−	0.436973i	0.975840	−	0.218487i	$-0.0701120\pi$
$432$		0			0
$433$	−36.9282			−1.77466			−0.887328	−	0.461139i	$-0.847441\pi$
$433$	−36.9282			−1.77466			−0.887328	+	0.461139i	$0.847441\pi$
$434$		0			0
$435$		−	3.46410i		−	0.166091i
$436$		0			0
$437$		−	70.6410i		−	3.37922i
$438$		0			0
$439$	−2.14359			−0.102308			−0.0511541	−	0.998691i	$-0.516290\pi$
$439$	−2.14359			−0.102308			−0.0511541	+	0.998691i	$0.516290\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$		0			0
$443$	−38.6410			−1.83589			−0.917945	−	0.396708i	$-0.870153\pi$
$443$	−38.6410			−1.83589			−0.917945	+	0.396708i	$0.870153\pi$
$444$		0			0
$445$	−0.928203			−0.0440011
$446$		0			0
$447$			0.928203i			0.0439025i
$448$		0			0
$449$		−	0.928203i		−	0.0438046i	−0.999760	−	0.0219023i	$-0.993028\pi$
$449$		−	0.928203i		−	0.0438046i	0.999760	−	0.0219023i	$-0.00697228\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			23.8564i			1.12087i
$454$		0			0
$455$	3.46410	+	12.0000i	0.162400	+	0.562569i
$456$		0			0
$457$		−	26.2487i		−	1.22786i	−0.789359	−	0.613931i	$-0.789587\pi$
$457$		−	26.2487i		−	1.22786i	0.789359	−	0.613931i	$-0.210413\pi$
$458$		0			0
$459$	−7.46410			−0.348394
$460$		0			0
$461$		−	0.928203i		−	0.0432307i	−0.999766	−	0.0216154i	$-0.993119\pi$
$461$		−	0.928203i		−	0.0432307i	0.999766	−	0.0216154i	$-0.00688092\pi$
$462$		0			0
$463$		−	27.1769i		−	1.26302i	−0.775368	−	0.631509i	$-0.782436\pi$
$463$		−	27.1769i		−	1.26302i	0.775368	−	0.631509i	$-0.217564\pi$
$464$		0			0
$465$	2.00000			0.0927478
$466$		0			0
$467$	25.8564			1.19649			0.598246	−	0.801313i	$-0.295865\pi$
$467$	25.8564			1.19649			0.598246	+	0.801313i	$0.295865\pi$
$468$		0			0
$469$	6.92820			0.319915
$470$		0			0
$471$	−7.85641			−0.362004
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	7.46410i		−	0.342476i
$476$		0			0
$477$	2.00000			0.0915737
$478$		0			0
$479$		−	18.9282i		−	0.864852i	−0.901670	−	0.432426i	$-0.857658\pi$
$479$		−	18.9282i		−	0.864852i	0.901670	−	0.432426i	$-0.142342\pi$
$480$		0			0
$481$	27.7128	−	8.00000i	1.26360	−	0.364769i
$482$		0			0
$483$		−	32.7846i		−	1.49175i
$484$		0			0
$485$	−6.53590			−0.296780
$486$		0			0
$487$		−	3.46410i		−	0.156973i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$		−	3.46410i		−	0.156973i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$		0			0
$489$		−	8.92820i		−	0.403747i
$490$		0			0
$491$	−31.3205			−1.41347			−0.706737	−	0.707476i	$-0.749834\pi$
$491$	−31.3205			−1.41347			−0.706737	+	0.707476i	$0.749834\pi$
$492$		0			0
$493$	−25.8564			−1.16451
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.1436			−0.455002
$498$		0			0
$499$			16.5359i			0.740248i	0.928982	+	0.370124i	$0.120685\pi$
$499$			16.5359i			0.740248i	−0.928982	+	0.370124i	$0.879315\pi$
$500$		0			0
$501$		−	9.85641i		−	0.440352i
$502$		0			0
$503$	−11.3205			−0.504757			−0.252378	−	0.967629i	$-0.581213\pi$
$503$	−11.3205			−0.504757			−0.252378	+	0.967629i	$0.581213\pi$
$504$		0			0
$505$			6.39230i			0.284454i
$506$		0			0
$507$	−11.0000	+	6.92820i	−0.488527	+	0.307692i
$508$		0			0
$509$			18.7846i			0.832613i	0.909224	+	0.416307i	$0.136676\pi$
$509$			18.7846i			0.832613i	−0.909224	+	0.416307i	$0.863324\pi$
$510$		0			0
$511$	−42.9282			−1.89903
$512$		0			0
$513$			7.46410i			0.329548i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−19.8564			−0.871600
$520$		0			0
$521$	30.7846			1.34870			0.674349	−	0.738413i	$-0.264424\pi$
$521$	30.7846			1.34870			0.674349	+	0.738413i	$0.264424\pi$
$522$		0			0
$523$	−23.7128			−1.03689			−0.518444	−	0.855111i	$-0.673489\pi$
$523$	−23.7128			−1.03689			−0.518444	+	0.855111i	$0.673489\pi$
$524$		0			0
$525$		−	3.46410i		−	0.151186i
$526$		0			0
$527$		−	14.9282i		−	0.650283i
$528$		0			0
$529$	66.5692			2.89431
$530$		0			0
$531$			14.9282i			0.647829i
$532$		0			0
$533$	−6.92820	+	2.00000i	−0.300094	+	0.0866296i
$534$		0			0
$535$		−	8.00000i		−	0.345870i
$536$		0			0
$537$	19.3205			0.833741
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	36.3923i		−	1.56463i	−0.622885	−	0.782314i	$-0.714039\pi$
$541$		−	36.3923i		−	1.56463i	0.622885	−	0.782314i	$-0.285961\pi$
$542$		0			0
$543$	−15.8564			−0.680464
$544$		0			0
$545$	9.46410			0.405398
$546$		0			0
$547$	−15.7128			−0.671831			−0.335916	−	0.941892i	$-0.609046\pi$
$547$	−15.7128			−0.671831			−0.335916	+	0.941892i	$0.609046\pi$
$548$		0			0
$549$	−8.92820			−0.381046
$550$		0			0
$551$			25.8564i			1.10152i
$552$		0			0
$553$			37.8564i			1.60982i
$554$		0			0
$555$	−8.00000			−0.339581
$556$		0			0
$557$			31.8564i			1.34980i	0.737910	+	0.674900i	$0.235813\pi$
$557$			31.8564i			1.34980i	−0.737910	+	0.674900i	$0.764187\pi$
$558$		0			0
$559$	6.92820	+	24.0000i	0.293032	+	1.01509i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.07180			0.213751			0.106875	−	0.994272i	$-0.465915\pi$
$563$	5.07180			0.213751			0.106875	+	0.994272i	$0.465915\pi$
$564$		0			0
$565$		−	8.53590i		−	0.359108i
$566$		0			0
$567$			3.46410i			0.145479i
$568$		0			0
$569$	4.92820			0.206601			0.103301	−	0.994650i	$-0.467060\pi$
$569$	4.92820			0.206601			0.103301	+	0.994650i	$0.467060\pi$
$570$		0			0
$571$	6.92820			0.289936			0.144968	−	0.989436i	$-0.453692\pi$
$571$	6.92820			0.289936			0.144968	+	0.989436i	$0.453692\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$		0			0
$575$	9.46410			0.394680
$576$		0			0
$577$			14.2487i			0.593182i	0.955005	+	0.296591i	$0.0958497\pi$
$577$			14.2487i			0.593182i	−0.955005	+	0.296591i	$0.904150\pi$
$578$		0			0
$579$		−	10.5359i		−	0.437857i
$580$		0			0
$581$	24.0000			0.995688
$582$		0			0
$583$		0			0
$584$		0			0
$585$	3.46410	−	1.00000i	0.143223	−	0.0413449i
$586$		0			0
$587$			34.6410i			1.42979i	0.699233	+	0.714894i	$0.253525\pi$
$587$			34.6410i			1.42979i	−0.699233	+	0.714894i	$0.746475\pi$
$588$		0			0
$589$	−14.9282			−0.615106
$590$		0			0
$591$		−	0.928203i		−	0.0381812i
$592$		0			0
$593$			12.9282i			0.530898i	0.964125	+	0.265449i	$0.0855201\pi$
$593$			12.9282i			0.530898i	−0.964125	+	0.265449i	$0.914480\pi$
$594$		0			0
$595$	−25.8564			−1.06001
$596$		0			0
$597$	16.7846			0.686948
$598$		0			0
$599$	37.8564			1.54677			0.773385	−	0.633936i	$-0.218562\pi$
$599$	37.8564			1.54677			0.773385	+	0.633936i	$0.218562\pi$
$600$		0			0
$601$	19.8564			0.809960			0.404980	−	0.914326i	$-0.367279\pi$
$601$	19.8564			0.809960			0.404980	+	0.914326i	$0.367279\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$		−	11.0000i		−	0.447214i
$606$		0			0
$607$	−0.784610			−0.0318463			−0.0159232	−	0.999873i	$-0.505069\pi$
$607$	−0.784610			−0.0318463			−0.0159232	+	0.999873i	$0.505069\pi$
$608$		0			0
$609$			12.0000i			0.486265i
$610$		0			0
$611$	13.8564	−	4.00000i	0.560570	−	0.161823i
$612$		0			0
$613$			27.7128i			1.11931i	0.828726	+	0.559655i	$0.189066\pi$
$613$			27.7128i			1.11931i	−0.828726	+	0.559655i	$0.810934\pi$
$614$		0			0
$615$	2.00000			0.0806478
$616$		0			0
$617$			2.78461i			0.112104i	0.998428	+	0.0560521i	$0.0178513\pi$
$617$			2.78461i			0.112104i	−0.998428	+	0.0560521i	$0.982149\pi$
$618$		0			0
$619$			19.4641i			0.782328i	0.920321	+	0.391164i	$0.127928\pi$
$619$			19.4641i			0.782328i	−0.920321	+	0.391164i	$0.872072\pi$
$620$		0			0
$621$	−9.46410			−0.379781
$622$		0			0
$623$	3.21539			0.128822
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			59.7128i			2.38091i
$630$		0			0
$631$		−	3.07180i		−	0.122286i	−0.998129	−	0.0611431i	$-0.980525\pi$
$631$		−	3.07180i		−	0.122286i	0.998129	−	0.0611431i	$-0.0194746\pi$
$632$		0			0
$633$	−6.92820			−0.275371
$634$		0			0
$635$		−	2.92820i		−	0.116202i
$636$		0			0
$637$	−5.00000	−	17.3205i	−0.198107	−	0.686264i
$638$		0			0
$639$			2.92820i			0.115838i
$640$		0			0
$641$	25.7128			1.01560			0.507798	−	0.861476i	$-0.330460\pi$
$641$	25.7128			1.01560			0.507798	+	0.861476i	$0.330460\pi$
$642$		0			0
$643$			38.0000i			1.49857i	0.662246	+	0.749287i	$0.269604\pi$
$643$			38.0000i			1.49857i	−0.662246	+	0.749287i	$0.730396\pi$
$644$		0			0
$645$		−	6.92820i		−	0.272798i
$646$		0			0
$647$	3.32051			0.130543			0.0652713	−	0.997868i	$-0.479209\pi$
$647$	3.32051			0.130543			0.0652713	+	0.997868i	$0.479209\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−6.92820			−0.271538
$652$		0			0
$653$	−23.0718			−0.902869			−0.451435	−	0.892304i	$-0.649088\pi$
$653$	−23.0718			−0.902869			−0.451435	+	0.892304i	$0.649088\pi$
$654$		0			0
$655$		−	5.46410i		−	0.213500i
$656$		0			0
$657$			12.3923i			0.483470i
$658$		0			0
$659$	17.4641			0.680305			0.340152	−	0.940370i	$-0.389521\pi$
$659$	17.4641			0.680305			0.340152	+	0.940370i	$0.389521\pi$
$660$		0			0
$661$		−	40.3923i		−	1.57108i	−0.618812	−	0.785539i	$-0.712386\pi$
$661$		−	40.3923i		−	1.57108i	0.618812	−	0.785539i	$-0.287614\pi$
$662$		0			0
$663$	−7.46410	−	25.8564i	−0.289882	−	1.00418i
$664$		0			0
$665$			25.8564i			1.00267i
$666$		0			0
$667$	−32.7846			−1.26943
$668$		0			0
$669$		−	14.3923i		−	0.556439i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	32.6410			1.25822			0.629109	−	0.777317i	$-0.283420\pi$
$673$	32.6410			1.25822			0.629109	+	0.777317i	$0.283420\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	−0.928203			−0.0356737			−0.0178369	−	0.999841i	$-0.505678\pi$
$677$	−0.928203			−0.0356737			−0.0178369	+	0.999841i	$0.505678\pi$
$678$		0			0
$679$	22.6410			0.868882
$680$		0			0
$681$			18.9282i			0.725330i
$682$		0			0
$683$			16.7846i			0.642245i	0.947038	+	0.321123i	$0.104060\pi$
$683$			16.7846i			0.642245i	−0.947038	+	0.321123i	$0.895940\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$		−	17.4641i		−	0.666297i
$688$		0			0
$689$	2.00000	+	6.92820i	0.0761939	+	0.263944i
$690$		0			0
$691$		−	12.5359i		−	0.476888i	−0.971156	−	0.238444i	$-0.923363\pi$
$691$		−	12.5359i		−	0.476888i	0.971156	−	0.238444i	$-0.0766374\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	14.9282i		−	0.566259i
$696$		0			0
$697$		−	14.9282i		−	0.565446i
$698$		0			0
$699$	26.3923			0.998249
$700$		0			0
$701$	13.6077			0.513956			0.256978	−	0.966417i	$-0.417273\pi$
$701$	13.6077			0.513956			0.256978	+	0.966417i	$0.417273\pi$
$702$		0			0
$703$	59.7128			2.25211
$704$		0			0
$705$	−4.00000			−0.150649
$706$		0			0
$707$		−	22.1436i		−	0.832796i
$708$		0			0
$709$		−	31.3205i		−	1.17627i	−0.808764	−	0.588133i	$-0.799863\pi$
$709$		−	31.3205i		−	1.17627i	0.808764	−	0.588133i	$-0.200137\pi$
$710$		0			0
$711$	10.9282			0.409840
$712$		0			0
$713$		−	18.9282i		−	0.708867i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			5.07180i			0.189410i
$718$		0			0
$719$	−50.6410			−1.88859			−0.944296	−	0.329098i	$-0.893255\pi$
$719$	−50.6410			−1.88859			−0.944296	+	0.329098i	$0.893255\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	17.8564i		−	0.664087i
$724$		0			0
$725$	−3.46410			−0.128654
$726$		0			0
$727$	−2.14359			−0.0795015			−0.0397507	−	0.999210i	$-0.512656\pi$
$727$	−2.14359			−0.0795015			−0.0397507	+	0.999210i	$0.512656\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−51.7128			−1.91267
$732$		0			0
$733$			9.85641i			0.364055i	0.983293	+	0.182027i	$0.0582659\pi$
$733$			9.85641i			0.364055i	−0.983293	+	0.182027i	$0.941734\pi$
$734$		0			0
$735$			5.00000i			0.184428i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			47.1769i			1.73543i	0.497061	+	0.867715i	$0.334412\pi$
$739$			47.1769i			1.73543i	−0.497061	+	0.867715i	$0.665588\pi$
$740$		0			0
$741$	−25.8564	+	7.46410i	−0.949859	+	0.274201i
$742$		0			0
$743$			46.6410i			1.71109i	0.517726	+	0.855546i	$0.326779\pi$
$743$			46.6410i			1.71109i	−0.517726	+	0.855546i	$0.673221\pi$
$744$		0			0
$745$	0.928203			0.0340067
$746$		0			0
$747$		−	6.92820i		−	0.253490i
$748$		0			0
$749$			27.7128i			1.01260i
$750$		0			0
$751$	−19.7128			−0.719331			−0.359665	−	0.933081i	$-0.617109\pi$
$751$	−19.7128			−0.719331			−0.359665	+	0.933081i	$0.617109\pi$
$752$		0			0
$753$	−14.5359			−0.529718
$754$		0			0
$755$	23.8564			0.868224
$756$		0			0
$757$	−41.7128			−1.51608			−0.758039	−	0.652209i	$-0.773842\pi$
$757$	−41.7128			−1.51608			−0.758039	+	0.652209i	$0.773842\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			49.7128i			1.80209i	0.433728	+	0.901044i	$0.357198\pi$
$761$			49.7128i			1.80209i	−0.433728	+	0.901044i	$0.642802\pi$
$762$		0			0
$763$	−32.7846			−1.18688
$764$		0			0
$765$			7.46410i			0.269865i
$766$		0			0
$767$	−51.7128	+	14.9282i	−1.86724	+	0.539026i
$768$		0			0
$769$			8.00000i			0.288487i	0.989542	+	0.144244i	$0.0460749\pi$
$769$			8.00000i			0.288487i	−0.989542	+	0.144244i	$0.953925\pi$
$770$		0			0
$771$	13.3205			0.479726
$772$		0			0
$773$			22.7846i			0.819505i	0.912197	+	0.409753i	$0.134385\pi$
$773$			22.7846i			0.819505i	−0.912197	+	0.409753i	$0.865615\pi$
$774$		0			0
$775$		−	2.00000i		−	0.0718421i
$776$		0			0
$777$	27.7128			0.994192
$778$		0			0
$779$	−14.9282			−0.534858
$780$		0			0
$781$		0			0
$782$		0			0
$783$	3.46410			0.123797
$784$		0			0
$785$			7.85641i			0.280407i
$786$		0			0
$787$			5.21539i			0.185909i	0.995670	+	0.0929543i	$0.0296310\pi$
$787$			5.21539i			0.185909i	−0.995670	+	0.0929543i	$0.970369\pi$
$788$		0			0
$789$	−5.46410			−0.194527
$790$		0			0
$791$			29.5692i			1.05136i
$792$		0			0
$793$	−8.92820	−	30.9282i	−0.317050	−	1.09829i
$794$		0			0
$795$		−	2.00000i		−	0.0709327i
$796$		0			0
$797$	11.8564			0.419975			0.209988	−	0.977704i	$-0.432658\pi$
$797$	11.8564			0.419975			0.209988	+	0.977704i	$0.432658\pi$
$798$		0			0
$799$			29.8564i			1.05624i
$800$		0			0
$801$		−	0.928203i		−	0.0327964i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−32.7846			−1.15551
$806$		0			0
$807$	8.53590			0.300478
$808$		0			0
$809$	−34.0000			−1.19538			−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000			−1.19538			−0.597688	+	0.801729i	$0.703914\pi$
$810$		0			0
$811$			35.1769i			1.23523i	0.786481	+	0.617614i	$0.211901\pi$
$811$			35.1769i			1.23523i	−0.786481	+	0.617614i	$0.788099\pi$
$812$		0			0
$813$			19.8564i			0.696395i
$814$		0			0
$815$	−8.92820			−0.312741
$816$		0			0
$817$			51.7128i			1.80920i
$818$		0			0
$819$	−12.0000	+	3.46410i	−0.419314	+	0.121046i
$820$		0			0
$821$			11.0718i			0.386408i	0.981159	+	0.193204i	$0.0618880\pi$
$821$			11.0718i			0.386408i	−0.981159	+	0.193204i	$0.938112\pi$
$822$		0			0
$823$	3.71281			0.129421			0.0647103	−	0.997904i	$-0.479388\pi$
$823$	3.71281			0.129421			0.0647103	+	0.997904i	$0.479388\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			24.7846i			0.861845i	0.902389	+	0.430923i	$0.141812\pi$
$827$			24.7846i			0.861845i	−0.902389	+	0.430923i	$0.858188\pi$
$828$		0			0
$829$	29.7128			1.03197			0.515984	−	0.856598i	$-0.327426\pi$
$829$	29.7128			1.03197			0.515984	+	0.856598i	$0.327426\pi$
$830$		0			0
$831$	−29.7128			−1.03073
$832$		0			0
$833$	37.3205			1.29308
$834$		0			0
$835$	−9.85641			−0.341095
$836$		0			0
$837$			2.00000i			0.0691301i
$838$		0			0
$839$		−	37.0718i		−	1.27986i	−0.768433	−	0.639930i	$-0.778963\pi$
$839$		−	37.0718i		−	1.27986i	0.768433	−	0.639930i	$-0.221037\pi$
$840$		0			0
$841$	−17.0000			−0.586207
$842$		0			0
$843$		−	4.92820i		−	0.169736i
$844$		0			0
$845$	6.92820	+	11.0000i	0.238337	+	0.378412i
$846$		0			0
$847$			38.1051i			1.30931i
$848$		0			0
$849$	22.9282			0.786894
$850$		0			0
$851$			75.7128i			2.59540i
$852$		0			0
$853$		−	18.1436i		−	0.621225i	−0.950537	−	0.310612i	$-0.899466\pi$
$853$		−	18.1436i		−	0.621225i	0.950537	−	0.310612i	$-0.100534\pi$
$854$		0			0
$855$	7.46410			0.255267
$856$		0			0
$857$	43.1769			1.47490			0.737448	−	0.675404i	$-0.236031\pi$
$857$	43.1769			1.47490			0.737448	+	0.675404i	$0.236031\pi$
$858$		0			0
$859$	−12.7846			−0.436205			−0.218103	−	0.975926i	$-0.569987\pi$
$859$	−12.7846			−0.436205			−0.218103	+	0.975926i	$0.569987\pi$
$860$		0			0
$861$	−6.92820			−0.236113
$862$		0			0
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$		0			0
$865$			19.8564i			0.675138i
$866$		0			0
$867$	38.7128			1.31476
$868$		0			0
$869$		0			0
$870$		0			0
$871$	6.92820	−	2.00000i	0.234753	−	0.0677674i
$872$		0			0
$873$		−	6.53590i		−	0.221207i
$874$		0			0
$875$	−3.46410			−0.117108
$876$		0			0
$877$		−	46.6410i		−	1.57496i	−0.616343	−	0.787478i	$-0.711387\pi$
$877$		−	46.6410i		−	1.57496i	0.616343	−	0.787478i	$-0.288613\pi$
$878$		0			0
$879$		−	24.9282i		−	0.840807i
$880$		0			0
$881$	−11.0718			−0.373018			−0.186509	−	0.982453i	$-0.559717\pi$
$881$	−11.0718			−0.373018			−0.186509	+	0.982453i	$0.559717\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$	14.9282			0.501806
$886$		0			0
$887$	−24.6795			−0.828656			−0.414328	−	0.910128i	$-0.635983\pi$
$887$	−24.6795			−0.828656			−0.414328	+	0.910128i	$0.635983\pi$
$888$		0			0
$889$			10.1436i			0.340205i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	29.8564			0.999107
$894$		0			0
$895$		−	19.3205i		−	0.645813i
$896$		0			0
$897$	−9.46410	−	32.7846i	−0.315997	−	1.09465i
$898$		0			0
$899$			6.92820i			0.231069i
$900$		0			0
$901$	−14.9282			−0.497331
$902$		0			0
$903$			24.0000i			0.798670i
$904$		0			0
$905$			15.8564i			0.527085i
$906$		0			0
$907$	−40.4974			−1.34469			−0.672347	−	0.740236i	$-0.734714\pi$
$907$	−40.4974			−1.34469			−0.672347	+	0.740236i	$0.734714\pi$
$908$		0			0
$909$	−6.39230			−0.212019
$910$		0			0
$911$	−46.9282			−1.55480			−0.777400	−	0.629006i	$-0.783462\pi$
$911$	−46.9282			−1.55480			−0.777400	+	0.629006i	$0.783462\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			8.92820i			0.295157i
$916$		0			0
$917$			18.9282i			0.625064i
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$		−	2.00000i		−	0.0659022i
$922$		0			0
$923$	−10.1436	+	2.92820i	−0.333880	+	0.0963830i
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	43.0718i		−	1.41314i	−0.707643	−	0.706570i	$-0.750242\pi$
$929$		−	43.0718i		−	1.41314i	0.707643	−	0.706570i	$-0.249758\pi$
$930$		0			0
$931$		−	37.3205i		−	1.22313i
$932$		0			0
$933$	−23.7128			−0.776323
$934$		0			0
$935$		0			0
$936$		0			0
$937$	35.0718			1.14575			0.572873	−	0.819644i	$-0.305829\pi$
$937$	35.0718			1.14575			0.572873	+	0.819644i	$0.305829\pi$
$938$		0			0
$939$	−15.8564			−0.517454
$940$		0			0
$941$			3.85641i			0.125715i	0.998023	+	0.0628576i	$0.0200214\pi$
$941$			3.85641i			0.125715i	−0.998023	+	0.0628576i	$0.979979\pi$
$942$		0			0
$943$		−	18.9282i		−	0.616387i
$944$		0			0
$945$	3.46410			0.112687
$946$		0			0
$947$			30.9282i			1.00503i	0.864568	+	0.502516i	$0.167592\pi$
$947$			30.9282i			1.00503i	−0.864568	+	0.502516i	$0.832408\pi$
$948$		0			0
$949$	−42.9282	+	12.3923i	−1.39351	+	0.402271i
$950$		0			0
$951$		−	11.8564i		−	0.384470i
$952$		0			0
$953$	−11.4641			−0.371359			−0.185679	−	0.982610i	$-0.559449\pi$
$953$	−11.4641			−0.371359			−0.185679	+	0.982610i	$0.559449\pi$
$954$		0			0
$955$		−	12.0000i		−	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−20.7846			−0.671170
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	8.00000			0.257796
$964$		0			0
$965$	−10.5359			−0.339163
$966$		0			0
$967$		−	27.4641i		−	0.883186i	−0.897215	−	0.441593i	$-0.854414\pi$
$967$		−	27.4641i		−	0.883186i	0.897215	−	0.441593i	$-0.145586\pi$
$968$		0			0
$969$		−	55.7128i		−	1.78975i
$970$		0			0
$971$	36.3923			1.16788			0.583942	−	0.811795i	$-0.301509\pi$
$971$	36.3923			1.16788			0.583942	+	0.811795i	$0.301509\pi$
$972$		0			0
$973$			51.7128i			1.65784i
$974$		0			0
$975$	−1.00000	−	3.46410i	−0.0320256	−	0.110940i
$976$		0			0
$977$			35.0718i			1.12205i	0.827800	+	0.561023i	$0.189592\pi$
$977$			35.0718i			1.12205i	−0.827800	+	0.561023i	$0.810408\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			9.46410i			0.302166i
$982$		0			0
$983$			26.9282i			0.858876i	0.903096	+	0.429438i	$0.141288\pi$
$983$			26.9282i			0.858876i	−0.903096	+	0.429438i	$0.858712\pi$
$984$		0			0
$985$	−0.928203			−0.0295750
$986$		0			0
$987$	13.8564			0.441054
$988$		0			0
$989$	−65.5692			−2.08498
$990$		0			0
$991$	30.6410			0.973344			0.486672	−	0.873585i	$-0.338211\pi$
$991$	30.6410			0.973344			0.486672	+	0.873585i	$0.338211\pi$
$992$		0			0
$993$		−	18.3923i		−	0.583662i
$994$		0			0
$995$		−	16.7846i		−	0.532108i
$996$		0			0
$997$	−21.7128			−0.687652			−0.343826	−	0.939033i	$-0.611723\pi$
$997$	−21.7128			−0.687652			−0.343826	+	0.939033i	$0.611723\pi$
$998$		0			0
$999$		−	8.00000i		−	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.g.g.961.2	✓	4
3.2	odd	2		4680.2.g.f.2521.4		4
4.3	odd	2		3120.2.g.m.961.1		4
13.12	even	2	inner	1560.2.g.g.961.3	yes	4
39.38	odd	2		4680.2.g.f.2521.1		4
52.51	odd	2		3120.2.g.m.961.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.g.g.961.2	✓	4	1.1	even	1	trivial
1560.2.g.g.961.3	yes	4	13.12	even	2	inner
3120.2.g.m.961.1		4	4.3	odd	2
3120.2.g.m.961.4		4	52.51	odd	2
4680.2.g.f.2521.1		4	39.38	odd	2
4680.2.g.f.2521.4		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} +3.46410i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} +3.46410i q^{7} +1.00000 q^{9} +(1.00000 + 3.46410i) q^{13} -1.00000i q^{15} -7.46410 q^{17} +7.46410i q^{19} +3.46410i q^{21} -9.46410 q^{23} -1.00000 q^{25} +1.00000 q^{27} +3.46410 q^{29} +2.00000i q^{31} +3.46410 q^{35} -8.00000i q^{37} +(1.00000 + 3.46410i) q^{39} +2.00000i q^{41} +6.92820 q^{43} -1.00000i q^{45} -4.00000i q^{47} -5.00000 q^{49} -7.46410 q^{51} +2.00000 q^{53} +7.46410i q^{57} +14.9282i q^{59} -8.92820 q^{61} +3.46410i q^{63} +(3.46410 - 1.00000i) q^{65} -2.00000i q^{67} -9.46410 q^{69} +2.92820i q^{71} +12.3923i q^{73} -1.00000 q^{75} +10.9282 q^{79} +1.00000 q^{81} -6.92820i q^{83} +7.46410i q^{85} +3.46410 q^{87} -0.928203i q^{89} +(-12.0000 + 3.46410i) q^{91} +2.00000i q^{93} +7.46410 q^{95} -6.53590i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{13} - 16 q^{17} - 24 q^{23} - 4 q^{25} + 4 q^{27} + 4 q^{39} - 20 q^{49} - 16 q^{51} + 8 q^{53} - 8 q^{61} - 24 q^{69} - 4 q^{75} + 16 q^{79} + 4 q^{81} - 48 q^{91} + 16 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^13 - 16 * q^17 - 24 * q^23 - 4 * q^25 + 4 * q^27 + 4 * q^39 - 20 * q^49 - 16 * q^51 + 8 * q^53 - 8 * q^61 - 24 * q^69 - 4 * q^75 + 16 * q^79 + 4 * q^81 - 48 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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