LMFDB - Embedded newform 4761.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.a (trivial)

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:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} +3.46410 q^{7} -1.73205 q^{8} +O(q^{10})$	$q+1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} +3.46410 q^{7} -1.73205 q^{8} +5.19615 q^{10} -5.00000 q^{13} +6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} -3.46410 q^{19} +3.00000 q^{20} +4.00000 q^{25} -8.66025 q^{26} +3.46410 q^{28} +1.73205 q^{29} +10.0000 q^{31} -5.19615 q^{32} +10.3923 q^{34} +10.3923 q^{35} -6.00000 q^{38} -5.19615 q^{40} +12.1244 q^{41} +3.46410 q^{43} +6.92820 q^{47} +5.00000 q^{49} +6.92820 q^{50} -5.00000 q^{52} +9.00000 q^{53} -6.00000 q^{56} +3.00000 q^{58} -10.3923 q^{59} +5.19615 q^{61} +17.3205 q^{62} +1.00000 q^{64} -15.0000 q^{65} +6.92820 q^{67} +6.00000 q^{68} +18.0000 q^{70} +3.46410 q^{71} +7.00000 q^{73} -3.46410 q^{76} -13.8564 q^{79} -15.0000 q^{80} +21.0000 q^{82} +6.00000 q^{83} +18.0000 q^{85} +6.00000 q^{86} -15.0000 q^{89} -17.3205 q^{91} +12.0000 q^{94} -10.3923 q^{95} -1.73205 q^{97} +8.66025 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{5}+O(q^{10}) $ 2 * q + 2 * q^4 + 6 * q^5	$ 2 q + 2 q^{4} + 6 q^{5} - 10 q^{13} + 12 q^{14} - 10 q^{16} + 12 q^{17} + 6 q^{20} + 8 q^{25} + 20 q^{31} - 12 q^{38} + 10 q^{49} - 10 q^{52} + 18 q^{53} - 12 q^{56} + 6 q^{58} + 2 q^{64} - 30 q^{65} + 12 q^{68} + 36 q^{70} + 14 q^{73} - 30 q^{80} + 42 q^{82} + 12 q^{83} + 36 q^{85} + 12 q^{86} - 30 q^{89} + 24 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^5 - 10 * q^13 + 12 * q^14 - 10 * q^16 + 12 * q^17 + 6 * q^20 + 8 * q^25 + 20 * q^31 - 12 * q^38 + 10 * q^49 - 10 * q^52 + 18 * q^53 - 12 * q^56 + 6 * q^58 + 2 * q^64 - 30 * q^65 + 12 * q^68 + 36 * q^70 + 14 * q^73 - 30 * q^80 + 42 * q^82 + 12 * q^83 + 36 * q^85 + 12 * q^86 - 30 * q^89 + 24 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	5.19615	1.64317
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−5.00000	−1.25000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.00000	0.800000
$26$	−8.66025	−1.69842
$27$	0	0
$28$	3.46410	0.654654
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	10.3923	1.78227
$35$	10.3923	1.75662
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−5.19615	−0.821584
$41$	12.1244	1.89351	0.946753	−	0.321960i	$-0.104342\pi$
$41$	12.1244	1.89351	0.946753	+	0.321960i	$0.104342\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	6.92820	0.979796
$51$	0	0
$52$	−5.00000	−0.693375
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	−6.00000	−0.801784
$57$	0	0
$58$	3.00000	0.393919
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	5.19615	0.665299	0.332650	−	0.943051i	$-0.392057\pi$
$61$	5.19615	0.665299	0.332650	+	0.943051i	$0.392057\pi$
$62$	17.3205	2.19971
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.0000	−1.86052
$66$	0	0
$67$	6.92820	0.846415	0.423207	−	0.906033i	$-0.360904\pi$
$67$	6.92820	0.846415	0.423207	+	0.906033i	$0.360904\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	18.0000	2.15141
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	0	0
$76$	−3.46410	−0.397360
$77$	0	0
$78$	0	0
$79$	−13.8564	−1.55897	−0.779484	−	0.626422i	$-0.784519\pi$
$79$	−13.8564	−1.55897	−0.779484	+	0.626422i	$0.784519\pi$
$80$	−15.0000	−1.67705
$81$	0	0
$82$	21.0000	2.31906
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	6.00000	0.646997
$87$	0	0
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	−17.3205	−1.81568
$92$	0	0
$93$	0	0
$94$	12.0000	1.23771
$95$	−10.3923	−1.06623
$96$	0	0
$97$	−1.73205	−0.175863	−0.0879316	−	0.996127i	$-0.528026\pi$
$97$	−1.73205	−0.175863	−0.0879316	+	0.996127i	$0.528026\pi$
$98$	8.66025	0.874818
$99$	0	0
$100$	4.00000	0.400000
$101$	−15.5885	−1.55111	−0.775555	−	0.631280i	$-0.782530\pi$
$101$	−15.5885	−1.55111	−0.775555	+	0.631280i	$0.782530\pi$
$102$	0	0
$103$	10.3923	1.02398	0.511992	−	0.858990i	$-0.328908\pi$
$103$	10.3923	1.02398	0.511992	+	0.858990i	$0.328908\pi$
$104$	8.66025	0.849208
$105$	0	0
$106$	15.5885	1.51408
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−15.5885	−1.49310	−0.746552	−	0.665327i	$-0.768292\pi$
$109$	−15.5885	−1.49310	−0.746552	+	0.665327i	$0.768292\pi$
$110$	0	0
$111$	0	0
$112$	−17.3205	−1.63663
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	0	0
$116$	1.73205	0.160817
$117$	0	0
$118$	−18.0000	−1.65703
$119$	20.7846	1.90532
$120$	0	0
$121$	−11.0000	−1.00000
$122$	9.00000	0.814822
$123$	0	0
$124$	10.0000	0.898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	−25.9808	−2.27866
$131$	−17.3205	−1.51330	−0.756650	−	0.653820i	$-0.773165\pi$
$131$	−17.3205	−1.51330	−0.756650	+	0.653820i	$0.773165\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	12.0000	1.03664
$135$	0	0
$136$	−10.3923	−0.891133
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	10.3923	0.878310
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	0	0
$145$	5.19615	0.431517
$146$	12.1244	1.00342
$147$	0	0
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	0	0
$155$	30.0000	2.40966
$156$	0	0
$157$	−5.19615	−0.414698	−0.207349	−	0.978267i	$-0.566484\pi$
$157$	−5.19615	−0.414698	−0.207349	+	0.978267i	$0.566484\pi$
$158$	−24.0000	−1.90934
$159$	0	0
$160$	−15.5885	−1.23238
$161$	0	0
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	12.1244	0.946753
$165$	0	0
$166$	10.3923	0.806599
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	31.1769	2.39116
$171$	0	0
$172$	3.46410	0.264135
$173$	5.19615	0.395056	0.197528	−	0.980297i	$-0.436709\pi$
$173$	5.19615	0.395056	0.197528	+	0.980297i	$0.436709\pi$
$174$	0	0
$175$	13.8564	1.04745
$176$	0	0
$177$	0	0
$178$	−25.9808	−1.94734
$179$	3.46410	0.258919	0.129460	−	0.991585i	$-0.458676\pi$
$179$	3.46410	0.258919	0.129460	+	0.991585i	$0.458676\pi$
$180$	0	0
$181$	−13.8564	−1.02994	−0.514969	−	0.857209i	$-0.672197\pi$
$181$	−13.8564	−1.02994	−0.514969	+	0.857209i	$0.672197\pi$
$182$	−30.0000	−2.22375
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	6.92820	0.505291
$189$	0	0
$190$	−18.0000	−1.30586
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	−3.00000	−0.215387
$195$	0	0
$196$	5.00000	0.357143
$197$	8.66025	0.617018	0.308509	−	0.951221i	$-0.400170\pi$
$197$	8.66025	0.617018	0.308509	+	0.951221i	$0.400170\pi$
$198$	0	0
$199$	−13.8564	−0.982255	−0.491127	−	0.871088i	$-0.663415\pi$
$199$	−13.8564	−0.982255	−0.491127	+	0.871088i	$0.663415\pi$
$200$	−6.92820	−0.489898
$201$	0	0
$202$	−27.0000	−1.89971
$203$	6.00000	0.421117
$204$	0	0
$205$	36.3731	2.54041
$206$	18.0000	1.25412
$207$	0	0
$208$	25.0000	1.73344
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	31.1769	2.13121
$215$	10.3923	0.708749
$216$	0	0
$217$	34.6410	2.35159
$218$	−27.0000	−1.82867
$219$	0	0
$220$	0	0
$221$	−30.0000	−2.01802
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−18.0000	−1.20268
$225$	0	0
$226$	−5.19615	−0.345643
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−19.0526	−1.24817	−0.624087	−	0.781355i	$-0.714529\pi$
$233$	−19.0526	−1.24817	−0.624087	+	0.781355i	$0.714529\pi$
$234$	0	0
$235$	20.7846	1.35584
$236$	−10.3923	−0.676481
$237$	0	0
$238$	36.0000	2.33353
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−5.19615	−0.334714	−0.167357	−	0.985896i	$-0.553523\pi$
$241$	−5.19615	−0.334714	−0.167357	+	0.985896i	$0.553523\pi$
$242$	−19.0526	−1.22474
$243$	0	0
$244$	5.19615	0.332650
$245$	15.0000	0.958315
$246$	0	0
$247$	17.3205	1.10208
$248$	−17.3205	−1.09985
$249$	0	0
$250$	−5.19615	−0.328634
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	−3.46410	−0.217357
$255$	0	0
$256$	19.0000	1.18750
$257$	8.66025	0.540212	0.270106	−	0.962831i	$-0.412941\pi$
$257$	8.66025	0.540212	0.270106	+	0.962831i	$0.412941\pi$
$258$	0	0
$259$	0	0
$260$	−15.0000	−0.930261
$261$	0	0
$262$	−30.0000	−1.85341
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	27.0000	1.65860
$266$	−20.7846	−1.27439
$267$	0	0
$268$	6.92820	0.423207
$269$	−13.8564	−0.844840	−0.422420	−	0.906400i	$-0.638819\pi$
$269$	−13.8564	−0.844840	−0.422420	+	0.906400i	$0.638819\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−30.0000	−1.81902
$273$	0	0
$274$	−5.19615	−0.313911
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−17.3205	−1.03882
$279$	0	0
$280$	−18.0000	−1.07571
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−24.2487	−1.44144	−0.720718	−	0.693228i	$-0.756188\pi$
$283$	−24.2487	−1.44144	−0.720718	+	0.693228i	$0.756188\pi$
$284$	3.46410	0.205557
$285$	0	0
$286$	0	0
$287$	42.0000	2.47918
$288$	0	0
$289$	19.0000	1.11765
$290$	9.00000	0.528498
$291$	0	0
$292$	7.00000	0.409644
$293$	−27.0000	−1.57736	−0.788678	−	0.614806i	$-0.789234\pi$
$293$	−27.0000	−1.57736	−0.788678	+	0.614806i	$0.789234\pi$
$294$	0	0
$295$	−31.1769	−1.81519
$296$	0	0
$297$	0	0
$298$	5.19615	0.301005
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	13.8564	0.797347
$303$	0	0
$304$	17.3205	0.993399
$305$	15.5885	0.892592
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	51.9615	2.95122
$311$	−3.46410	−0.196431	−0.0982156	−	0.995165i	$-0.531313\pi$
$311$	−3.46410	−0.196431	−0.0982156	+	0.995165i	$0.531313\pi$
$312$	0	0
$313$	−5.19615	−0.293704	−0.146852	−	0.989158i	$-0.546914\pi$
$313$	−5.19615	−0.293704	−0.146852	+	0.989158i	$0.546914\pi$
$314$	−9.00000	−0.507899
$315$	0	0
$316$	−13.8564	−0.779484
$317$	5.19615	0.291845	0.145922	−	0.989296i	$-0.453385\pi$
$317$	5.19615	0.291845	0.145922	+	0.989296i	$0.453385\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	−20.7846	−1.15649
$324$	0	0
$325$	−20.0000	−1.10940
$326$	−24.2487	−1.34301
$327$	0	0
$328$	−21.0000	−1.15953
$329$	24.0000	1.32316
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−18.0000	−0.984916
$335$	20.7846	1.13558
$336$	0	0
$337$	8.66025	0.471754	0.235877	−	0.971783i	$-0.424204\pi$
$337$	8.66025	0.471754	0.235877	+	0.971783i	$0.424204\pi$
$338$	20.7846	1.13053
$339$	0	0
$340$	18.0000	0.976187
$341$	0	0
$342$	0	0
$343$	−6.92820	−0.374088
$344$	−6.00000	−0.323498
$345$	0	0
$346$	9.00000	0.483843
$347$	−20.7846	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	24.0000	1.28285
$351$	0	0
$352$	0	0
$353$	−19.0526	−1.01407	−0.507033	−	0.861927i	$-0.669258\pi$
$353$	−19.0526	−1.01407	−0.507033	+	0.861927i	$0.669258\pi$
$354$	0	0
$355$	10.3923	0.551566
$356$	−15.0000	−0.794998
$357$	0	0
$358$	6.00000	0.317110
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	−24.0000	−1.26141
$363$	0	0
$364$	−17.3205	−0.907841
$365$	21.0000	1.09919
$366$	0	0
$367$	−6.92820	−0.361649	−0.180825	−	0.983515i	$-0.557877\pi$
$367$	−6.92820	−0.361649	−0.180825	+	0.983515i	$0.557877\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.1769	1.61862
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	−8.66025	−0.446026
$378$	0	0
$379$	10.3923	0.533817	0.266908	−	0.963722i	$-0.413998\pi$
$379$	10.3923	0.533817	0.266908	+	0.963722i	$0.413998\pi$
$380$	−10.3923	−0.533114
$381$	0	0
$382$	−10.3923	−0.531717
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	0	0
$386$	29.4449	1.49870
$387$	0	0
$388$	−1.73205	−0.0879316
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	−8.66025	−0.437409
$393$	0	0
$394$	15.0000	0.755689
$395$	−41.5692	−2.09157
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	−20.0000	−1.00000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	−50.0000	−2.49068
$404$	−15.5885	−0.775555
$405$	0	0
$406$	10.3923	0.515761
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	63.0000	3.11135
$411$	0	0
$412$	10.3923	0.511992
$413$	−36.0000	−1.77144
$414$	0	0
$415$	18.0000	0.883585
$416$	25.9808	1.27381
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−13.8564	−0.675320	−0.337660	−	0.941268i	$-0.609635\pi$
$421$	−13.8564	−0.675320	−0.337660	+	0.941268i	$0.609635\pi$
$422$	6.92820	0.337260
$423$	0	0
$424$	−15.5885	−0.757042
$425$	24.0000	1.16417
$426$	0	0
$427$	18.0000	0.871081
$428$	18.0000	0.870063
$429$	0	0
$430$	18.0000	0.868037
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	32.9090	1.58150	0.790752	−	0.612137i	$-0.209690\pi$
$433$	32.9090	1.58150	0.790752	+	0.612137i	$0.209690\pi$
$434$	60.0000	2.88009
$435$	0	0
$436$	−15.5885	−0.746552
$437$	0	0
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	−51.9615	−2.47156
$443$	−17.3205	−0.822922	−0.411461	−	0.911427i	$-0.634981\pi$
$443$	−17.3205	−0.822922	−0.411461	+	0.911427i	$0.634981\pi$
$444$	0	0
$445$	−45.0000	−2.13320
$446$	−6.92820	−0.328060
$447$	0	0
$448$	3.46410	0.163663
$449$	−15.5885	−0.735665	−0.367832	−	0.929892i	$-0.619900\pi$
$449$	−15.5885	−0.735665	−0.367832	+	0.929892i	$0.619900\pi$
$450$	0	0
$451$	0	0
$452$	−3.00000	−0.141108
$453$	0	0
$454$	0	0
$455$	−51.9615	−2.43599
$456$	0	0
$457$	1.73205	0.0810219	0.0405110	−	0.999179i	$-0.487101\pi$
$457$	1.73205	0.0810219	0.0405110	+	0.999179i	$0.487101\pi$
$458$	12.0000	0.560723
$459$	0	0
$460$	0	0
$461$	8.66025	0.403348	0.201674	−	0.979453i	$-0.435362\pi$
$461$	8.66025	0.403348	0.201674	+	0.979453i	$0.435362\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−8.66025	−0.402042
$465$	0	0
$466$	−33.0000	−1.52870
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	36.0000	1.66056
$471$	0	0
$472$	18.0000	0.828517
$473$	0	0
$474$	0	0
$475$	−13.8564	−0.635776
$476$	20.7846	0.952661
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	0	0
$482$	−9.00000	−0.409939
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−5.19615	−0.235945
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−9.00000	−0.407411
$489$	0	0
$490$	25.9808	1.17369
$491$	−38.1051	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	−38.1051	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$492$	0	0
$493$	10.3923	0.468046
$494$	30.0000	1.34976
$495$	0	0
$496$	−50.0000	−2.24507
$497$	12.0000	0.538274
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	31.1769	1.39149
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−46.7654	−2.08103
$506$	0	0
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	−22.5167	−0.998033	−0.499017	−	0.866592i	$-0.666305\pi$
$509$	−22.5167	−0.998033	−0.499017	+	0.866592i	$0.666305\pi$
$510$	0	0
$511$	24.2487	1.07270
$512$	8.66025	0.382733
$513$	0	0
$514$	15.0000	0.661622
$515$	31.1769	1.37382
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	25.9808	1.13933
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−6.92820	−0.302949	−0.151475	−	0.988461i	$-0.548402\pi$
$523$	−6.92820	−0.302949	−0.151475	+	0.988461i	$0.548402\pi$
$524$	−17.3205	−0.756650
$525$	0	0
$526$	−31.1769	−1.35938
$527$	60.0000	2.61364
$528$	0	0
$529$	0	0
$530$	46.7654	2.03136
$531$	0	0
$532$	−12.0000	−0.520266
$533$	−60.6218	−2.62582
$534$	0	0
$535$	54.0000	2.33462
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−24.0000	−1.03471
$539$	0	0
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	27.7128	1.19037
$543$	0	0
$544$	−31.1769	−1.33670
$545$	−46.7654	−2.00321
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−3.00000	−0.128154
$549$	0	0
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	−48.0000	−2.04117
$554$	17.3205	0.735878
$555$	0	0
$556$	−10.0000	−0.424094
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	0	0
$559$	−17.3205	−0.732579
$560$	−51.9615	−2.19578
$561$	0	0
$562$	−51.9615	−2.19186
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	−42.0000	−1.76539
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	0	0
$571$	−20.7846	−0.869809	−0.434904	−	0.900477i	$-0.643218\pi$
$571$	−20.7846	−0.869809	−0.434904	+	0.900477i	$0.643218\pi$
$572$	0	0
$573$	0	0
$574$	72.7461	3.03636
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	32.9090	1.36883
$579$	0	0
$580$	5.19615	0.215758
$581$	20.7846	0.862291
$582$	0	0
$583$	0	0
$584$	−12.1244	−0.501709
$585$	0	0
$586$	−46.7654	−1.93186
$587$	34.6410	1.42979	0.714894	−	0.699233i	$-0.246475\pi$
$587$	34.6410	1.42979	0.714894	+	0.699233i	$0.246475\pi$
$588$	0	0
$589$	−34.6410	−1.42736
$590$	−54.0000	−2.22314
$591$	0	0
$592$	0	0
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	0	0
$595$	62.3538	2.55626
$596$	3.00000	0.122885
$597$	0	0
$598$	0	0
$599$	−10.3923	−0.424618	−0.212309	−	0.977203i	$-0.568098\pi$
$599$	−10.3923	−0.424618	−0.212309	+	0.977203i	$0.568098\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	20.7846	0.847117
$603$	0	0
$604$	8.00000	0.325515
$605$	−33.0000	−1.34164
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	18.0000	0.729996
$609$	0	0
$610$	27.0000	1.09320
$611$	−34.6410	−1.40143
$612$	0	0
$613$	25.9808	1.04935	0.524677	−	0.851302i	$-0.324186\pi$
$613$	25.9808	1.04935	0.524677	+	0.851302i	$0.324186\pi$
$614$	48.4974	1.95720
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−13.8564	−0.556936	−0.278468	−	0.960446i	$-0.589827\pi$
$619$	−13.8564	−0.556936	−0.278468	+	0.960446i	$0.589827\pi$
$620$	30.0000	1.20483
$621$	0	0
$622$	−6.00000	−0.240578
$623$	−51.9615	−2.08179
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−9.00000	−0.359712
$627$	0	0
$628$	−5.19615	−0.207349
$629$	0	0
$630$	0	0
$631$	−17.3205	−0.689519	−0.344759	−	0.938691i	$-0.612039\pi$
$631$	−17.3205	−0.689519	−0.344759	+	0.938691i	$0.612039\pi$
$632$	24.0000	0.954669
$633$	0	0
$634$	9.00000	0.357436
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−25.0000	−0.990536
$638$	0	0
$639$	0	0
$640$	36.3731	1.43777
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−17.3205	−0.683054	−0.341527	−	0.939872i	$-0.610944\pi$
$643$	−17.3205	−0.683054	−0.341527	+	0.939872i	$0.610944\pi$
$644$	0	0
$645$	0	0
$646$	−36.0000	−1.41640
$647$	27.7128	1.08950	0.544752	−	0.838597i	$-0.316624\pi$
$647$	27.7128	1.08950	0.544752	+	0.838597i	$0.316624\pi$
$648$	0	0
$649$	0	0
$650$	−34.6410	−1.35873
$651$	0	0
$652$	−14.0000	−0.548282
$653$	8.66025	0.338902	0.169451	−	0.985539i	$-0.445801\pi$
$653$	8.66025	0.338902	0.169451	+	0.985539i	$0.445801\pi$
$654$	0	0
$655$	−51.9615	−2.03030
$656$	−60.6218	−2.36688
$657$	0	0
$658$	41.5692	1.62054
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−22.5167	−0.875797	−0.437898	−	0.899025i	$-0.644277\pi$
$661$	−22.5167	−0.875797	−0.437898	+	0.899025i	$0.644277\pi$
$662$	17.3205	0.673181
$663$	0	0
$664$	−10.3923	−0.403300
$665$	−36.0000	−1.39602
$666$	0	0
$667$	0	0
$668$	−10.3923	−0.402090
$669$	0	0
$670$	36.0000	1.39080
$671$	0	0
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	15.0000	0.577778
$675$	0	0
$676$	12.0000	0.461538
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	−31.1769	−1.19558
$681$	0	0
$682$	0	0
$683$	27.7128	1.06040	0.530201	−	0.847872i	$-0.322117\pi$
$683$	27.7128	1.06040	0.530201	+	0.847872i	$0.322117\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	−12.0000	−0.458162
$687$	0	0
$688$	−17.3205	−0.660338
$689$	−45.0000	−1.71436
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	5.19615	0.197528
$693$	0	0
$694$	−36.0000	−1.36654
$695$	−30.0000	−1.13796
$696$	0	0
$697$	72.7461	2.75546
$698$	45.0333	1.70454
$699$	0	0
$700$	13.8564	0.523723
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−33.0000	−1.24197
$707$	−54.0000	−2.03088
$708$	0	0
$709$	−19.0526	−0.715534	−0.357767	−	0.933811i	$-0.616462\pi$
$709$	−19.0526	−0.715534	−0.357767	+	0.933811i	$0.616462\pi$
$710$	18.0000	0.675528
$711$	0	0
$712$	25.9808	0.973670
$713$	0	0
$714$	0	0
$715$	0	0
$716$	3.46410	0.129460
$717$	0	0
$718$	−10.3923	−0.387837
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	36.0000	1.34071
$722$	−12.1244	−0.451222
$723$	0	0
$724$	−13.8564	−0.514969
$725$	6.92820	0.257307
$726$	0	0
$727$	−31.1769	−1.15629	−0.578144	−	0.815935i	$-0.696223\pi$
$727$	−31.1769	−1.15629	−0.578144	+	0.815935i	$0.696223\pi$
$728$	30.0000	1.11187
$729$	0	0
$730$	36.3731	1.34623
$731$	20.7846	0.768747
$732$	0	0
$733$	19.0526	0.703722	0.351861	−	0.936052i	$-0.385549\pi$
$733$	19.0526	0.703722	0.351861	+	0.936052i	$0.385549\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	0	0
$742$	54.0000	1.98240
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	0	0
$748$	0	0
$749$	62.3538	2.27836
$750$	0	0
$751$	34.6410	1.26407	0.632034	−	0.774940i	$-0.282220\pi$
$751$	34.6410	1.26407	0.632034	+	0.774940i	$0.282220\pi$
$752$	−34.6410	−1.26323
$753$	0	0
$754$	−15.0000	−0.546268
$755$	24.0000	0.873449
$756$	0	0
$757$	1.73205	0.0629525	0.0314762	−	0.999505i	$-0.489979\pi$
$757$	1.73205	0.0629525	0.0314762	+	0.999505i	$0.489979\pi$
$758$	18.0000	0.653789
$759$	0	0
$760$	18.0000	0.652929
$761$	−29.4449	−1.06738	−0.533688	−	0.845682i	$-0.679194\pi$
$761$	−29.4449	−1.06738	−0.533688	+	0.845682i	$0.679194\pi$
$762$	0	0
$763$	−54.0000	−1.95493
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−51.9615	−1.87745
$767$	51.9615	1.87622
$768$	0	0
$769$	34.6410	1.24919	0.624593	−	0.780950i	$-0.285265\pi$
$769$	34.6410	1.24919	0.624593	+	0.780950i	$0.285265\pi$
$770$	0	0
$771$	0	0
$772$	17.0000	0.611843
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	3.00000	0.107694
$777$	0	0
$778$	10.3923	0.372582
$779$	−42.0000	−1.50481
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−25.0000	−0.892857
$785$	−15.5885	−0.556376
$786$	0	0
$787$	−34.6410	−1.23482	−0.617409	−	0.786642i	$-0.711818\pi$
$787$	−34.6410	−1.23482	−0.617409	+	0.786642i	$0.711818\pi$
$788$	8.66025	0.308509
$789$	0	0
$790$	−72.0000	−2.56165
$791$	−10.3923	−0.369508
$792$	0	0
$793$	−25.9808	−0.922604
$794$	−12.1244	−0.430277
$795$	0	0
$796$	−13.8564	−0.491127
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	41.5692	1.47061
$800$	−20.7846	−0.734847
$801$	0	0
$802$	46.7654	1.65134
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−86.6025	−3.05044
$807$	0	0
$808$	27.0000	0.949857
$809$	−20.7846	−0.730748	−0.365374	−	0.930861i	$-0.619059\pi$
$809$	−20.7846	−0.730748	−0.365374	+	0.930861i	$0.619059\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	0	0
$815$	−42.0000	−1.47120
$816$	0	0
$817$	−12.0000	−0.419827
$818$	24.2487	0.847836
$819$	0	0
$820$	36.3731	1.27020
$821$	−15.5885	−0.544041	−0.272020	−	0.962291i	$-0.587692\pi$
$821$	−15.5885	−0.544041	−0.272020	+	0.962291i	$0.587692\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−18.0000	−0.627060
$825$	0	0
$826$	−62.3538	−2.16957
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	0	0
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	31.1769	1.08217
$831$	0	0
$832$	−5.00000	−0.173344
$833$	30.0000	1.03944
$834$	0	0
$835$	−31.1769	−1.07892
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−24.0000	−0.827095
$843$	0	0
$844$	4.00000	0.137686
$845$	36.0000	1.23844
$846$	0	0
$847$	−38.1051	−1.30931
$848$	−45.0000	−1.54531
$849$	0	0
$850$	41.5692	1.42581
$851$	0	0
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	31.1769	1.06685
$855$	0	0
$856$	−31.1769	−1.06561
$857$	−1.73205	−0.0591657	−0.0295829	−	0.999562i	$-0.509418\pi$
$857$	−1.73205	−0.0591657	−0.0295829	+	0.999562i	$0.509418\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	10.3923	0.354375
$861$	0	0
$862$	41.5692	1.41585
$863$	−6.92820	−0.235839	−0.117919	−	0.993023i	$-0.537622\pi$
$863$	−6.92820	−0.235839	−0.117919	+	0.993023i	$0.537622\pi$
$864$	0	0
$865$	15.5885	0.530023
$866$	57.0000	1.93694
$867$	0	0
$868$	34.6410	1.17579
$869$	0	0
$870$	0	0
$871$	−34.6410	−1.17377
$872$	27.0000	0.914335
$873$	0	0
$874$	0	0
$875$	−10.3923	−0.351324
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	45.0333	1.51980
$879$	0	0
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	−30.0000	−1.00901
$885$	0	0
$886$	−30.0000	−1.00787
$887$	−34.6410	−1.16313	−0.581566	−	0.813499i	$-0.697560\pi$
$887$	−34.6410	−1.16313	−0.581566	+	0.813499i	$0.697560\pi$
$888$	0	0
$889$	−6.92820	−0.232364
$890$	−77.9423	−2.61263
$891$	0	0
$892$	−4.00000	−0.133930
$893$	−24.0000	−0.803129
$894$	0	0
$895$	10.3923	0.347376
$896$	42.0000	1.40312
$897$	0	0
$898$	−27.0000	−0.901002
$899$	17.3205	0.577671
$900$	0	0
$901$	54.0000	1.79900
$902$	0	0
$903$	0	0
$904$	5.19615	0.172821
$905$	−41.5692	−1.38181
$906$	0	0
$907$	−20.7846	−0.690142	−0.345071	−	0.938577i	$-0.612145\pi$
$907$	−20.7846	−0.690142	−0.345071	+	0.938577i	$0.612145\pi$
$908$	0	0
$909$	0	0
$910$	−90.0000	−2.98347
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	0	0
$914$	3.00000	0.0992312
$915$	0	0
$916$	6.92820	0.228914
$917$	−60.0000	−1.98137
$918$	0	0
$919$	3.46410	0.114270	0.0571351	−	0.998366i	$-0.481803\pi$
$919$	3.46410	0.114270	0.0571351	+	0.998366i	$0.481803\pi$
$920$	0	0
$921$	0	0
$922$	15.0000	0.493999
$923$	−17.3205	−0.570111
$924$	0	0
$925$	0	0
$926$	−6.92820	−0.227675
$927$	0	0
$928$	−9.00000	−0.295439
$929$	48.4974	1.59115	0.795574	−	0.605856i	$-0.207169\pi$
$929$	48.4974	1.59115	0.795574	+	0.605856i	$0.207169\pi$
$930$	0	0
$931$	−17.3205	−0.567657
$932$	−19.0526	−0.624087
$933$	0	0
$934$	−20.7846	−0.680093
$935$	0	0
$936$	0	0
$937$	20.7846	0.679004	0.339502	−	0.940605i	$-0.389742\pi$
$937$	20.7846	0.679004	0.339502	+	0.940605i	$0.389742\pi$
$938$	41.5692	1.35728
$939$	0	0
$940$	20.7846	0.677919
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	0	0
$944$	51.9615	1.69120
$945$	0	0
$946$	0	0
$947$	27.7128	0.900545	0.450273	−	0.892891i	$-0.351327\pi$
$947$	27.7128	0.900545	0.450273	+	0.892891i	$0.351327\pi$
$948$	0	0
$949$	−35.0000	−1.13615
$950$	−24.0000	−0.778663
$951$	0	0
$952$	−36.0000	−1.16677
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	−6.92820	−0.224074
$957$	0	0
$958$	−31.1769	−1.00728
$959$	−10.3923	−0.335585
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	−5.19615	−0.167357
$965$	51.0000	1.64175
$966$	0	0
$967$	−26.0000	−0.836104	−0.418052	−	0.908423i	$-0.637287\pi$
$967$	−26.0000	−0.836104	−0.418052	+	0.908423i	$0.637287\pi$
$968$	19.0526	0.612372
$969$	0	0
$970$	−9.00000	−0.288973
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	−34.6410	−1.11054
$974$	65.8179	2.10894
$975$	0	0
$976$	−25.9808	−0.831624
$977$	−39.0000	−1.24772	−0.623860	−	0.781536i	$-0.714437\pi$
$977$	−39.0000	−1.24772	−0.623860	+	0.781536i	$0.714437\pi$
$978$	0	0
$979$	0	0
$980$	15.0000	0.479157
$981$	0	0
$982$	−66.0000	−2.10614
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	25.9808	0.827816
$986$	18.0000	0.573237
$987$	0	0
$988$	17.3205	0.551039
$989$	0	0
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	−51.9615	−1.64978
$993$	0	0
$994$	20.7846	0.659248
$995$	−41.5692	−1.31783
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	6.92820	0.219308
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.u.1.2	yes	2
3.2	odd	2		4761.2.a.r.1.1	✓	2
23.22	odd	2		4761.2.a.r.1.2	yes	2
69.68	even	2	inner	4761.2.a.u.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4761.2.a.r.1.1	✓	2	3.2	odd	2
4761.2.a.r.1.2	yes	2	23.22	odd	2
4761.2.a.u.1.1	yes	2	69.68	even	2	inner
4761.2.a.u.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} +3.46410 q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} +3.46410 q^{7} -1.73205 q^{8} +5.19615 q^{10} -5.00000 q^{13} +6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} -3.46410 q^{19} +3.00000 q^{20} +4.00000 q^{25} -8.66025 q^{26} +3.46410 q^{28} +1.73205 q^{29} +10.0000 q^{31} -5.19615 q^{32} +10.3923 q^{34} +10.3923 q^{35} -6.00000 q^{38} -5.19615 q^{40} +12.1244 q^{41} +3.46410 q^{43} +6.92820 q^{47} +5.00000 q^{49} +6.92820 q^{50} -5.00000 q^{52} +9.00000 q^{53} -6.00000 q^{56} +3.00000 q^{58} -10.3923 q^{59} +5.19615 q^{61} +17.3205 q^{62} +1.00000 q^{64} -15.0000 q^{65} +6.92820 q^{67} +6.00000 q^{68} +18.0000 q^{70} +3.46410 q^{71} +7.00000 q^{73} -3.46410 q^{76} -13.8564 q^{79} -15.0000 q^{80} +21.0000 q^{82} +6.00000 q^{83} +18.0000 q^{85} +6.00000 q^{86} -15.0000 q^{89} -17.3205 q^{91} +12.0000 q^{94} -10.3923 q^{95} -1.73205 q^{97} +8.66025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{5}+O(q^{10}) \) 2 * q + 2 * q^4 + 6 * q^5	\( 2 q + 2 q^{4} + 6 q^{5} - 10 q^{13} + 12 q^{14} - 10 q^{16} + 12 q^{17} + 6 q^{20} + 8 q^{25} + 20 q^{31} - 12 q^{38} + 10 q^{49} - 10 q^{52} + 18 q^{53} - 12 q^{56} + 6 q^{58} + 2 q^{64} - 30 q^{65} + 12 q^{68} + 36 q^{70} + 14 q^{73} - 30 q^{80} + 42 q^{82} + 12 q^{83} + 36 q^{85} + 12 q^{86} - 30 q^{89} + 24 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^5 - 10 * q^13 + 12 * q^14 - 10 * q^16 + 12 * q^17 + 6 * q^20 + 8 * q^25 + 20 * q^31 - 12 * q^38 + 10 * q^49 - 10 * q^52 + 18 * q^53 - 12 * q^56 + 6 * q^58 + 2 * q^64 - 30 * q^65 + 12 * q^68 + 36 * q^70 + 14 * q^73 - 30 * q^80 + 42 * q^82 + 12 * q^83 + 36 * q^85 + 12 * q^86 - 30 * q^89 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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