LMFDB - Embedded newform 6664.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6664,2,Mod(1,6664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6664, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.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:	$53.2123079070$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{3} +2.61803 q^{5} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} +2.61803 q^{5} -0.381966 q^{9} +4.47214 q^{11} -4.47214 q^{13} +4.23607 q^{15} -1.00000 q^{17} +6.47214 q^{19} -4.76393 q^{23} +1.85410 q^{25} -5.47214 q^{27} +2.00000 q^{29} +9.09017 q^{31} +7.23607 q^{33} +4.00000 q^{37} -7.23607 q^{39} +10.0902 q^{41} -10.0902 q^{43} -1.00000 q^{45} +11.2361 q^{47} -1.61803 q^{51} -3.09017 q^{53} +11.7082 q^{55} +10.4721 q^{57} +4.00000 q^{59} -7.61803 q^{61} -11.7082 q^{65} +11.0902 q^{67} -7.70820 q^{69} -8.56231 q^{73} +3.00000 q^{75} +8.18034 q^{79} -7.70820 q^{81} +11.2361 q^{83} -2.61803 q^{85} +3.23607 q^{87} -1.70820 q^{89} +14.7082 q^{93} +16.9443 q^{95} +14.0344 q^{97} -1.70820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 3 q^{5} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 3 * q^5 - 3 * q^9	$ 2 q + q^{3} + 3 q^{5} - 3 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{19} - 14 q^{23} - 3 q^{25} - 2 q^{27} + 4 q^{29} + 7 q^{31} + 10 q^{33} + 8 q^{37} - 10 q^{39} + 9 q^{41} - 9 q^{43} - 2 q^{45} + 18 q^{47} - q^{51} + 5 q^{53} + 10 q^{55} + 12 q^{57} + 8 q^{59} - 13 q^{61} - 10 q^{65} + 11 q^{67} - 2 q^{69} + 3 q^{73} + 6 q^{75} - 6 q^{79} - 2 q^{81} + 18 q^{83} - 3 q^{85} + 2 q^{87} + 10 q^{89} + 16 q^{93} + 16 q^{95} - q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q + q^3 + 3 * q^5 - 3 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^19 - 14 * q^23 - 3 * q^25 - 2 * q^27 + 4 * q^29 + 7 * q^31 + 10 * q^33 + 8 * q^37 - 10 * q^39 + 9 * q^41 - 9 * q^43 - 2 * q^45 + 18 * q^47 - q^51 + 5 * q^53 + 10 * q^55 + 12 * q^57 + 8 * q^59 - 13 * q^61 - 10 * q^65 + 11 * q^67 - 2 * q^69 + 3 * q^73 + 6 * q^75 - 6 * q^79 - 2 * q^81 + 18 * q^83 - 3 * q^85 + 2 * q^87 + 10 * q^89 + 16 * q^93 + 16 * q^95 - q^97 + 10 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	2.61803	1.17082	0.585410	−	0.810737i	$-0.300933\pi$
$5$	2.61803	1.17082	0.585410	+	0.810737i	$0.300933\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	4.23607	1.09375
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.76393	−0.993348	−0.496674	−	0.867937i	$-0.665446\pi$
$23$	−4.76393	−0.993348	−0.496674	+	0.867937i	$0.665446\pi$
$24$	0	0
$25$	1.85410	0.370820
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	9.09017	1.63264	0.816321	−	0.577598i	$-0.196010\pi$
$31$	9.09017	1.63264	0.816321	+	0.577598i	$0.196010\pi$
$32$	0	0
$33$	7.23607	1.25964
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−7.23607	−1.15870
$40$	0	0
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	0	0
$43$	−10.0902	−1.53874	−0.769368	−	0.638806i	$-0.779429\pi$
$43$	−10.0902	−1.53874	−0.769368	+	0.638806i	$0.779429\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	11.2361	1.63895	0.819474	−	0.573116i	$-0.194265\pi$
$47$	11.2361	1.63895	0.819474	+	0.573116i	$0.194265\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.61803	−0.226570
$52$	0	0
$53$	−3.09017	−0.424467	−0.212234	−	0.977219i	$-0.568074\pi$
$53$	−3.09017	−0.424467	−0.212234	+	0.977219i	$0.568074\pi$
$54$	0	0
$55$	11.7082	1.57873
$56$	0	0
$57$	10.4721	1.38707
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−7.61803	−0.975389	−0.487695	−	0.873014i	$-0.662162\pi$
$61$	−7.61803	−0.975389	−0.487695	+	0.873014i	$0.662162\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.7082	−1.45222
$66$	0	0
$67$	11.0902	1.35488	0.677440	−	0.735578i	$-0.263089\pi$
$67$	11.0902	1.35488	0.677440	+	0.735578i	$0.263089\pi$
$68$	0	0
$69$	−7.70820	−0.927959
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−8.56231	−1.00214	−0.501071	−	0.865406i	$-0.667060\pi$
$73$	−8.56231	−1.00214	−0.501071	+	0.865406i	$0.667060\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.18034	0.920360	0.460180	−	0.887826i	$-0.347785\pi$
$79$	8.18034	0.920360	0.460180	+	0.887826i	$0.347785\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	11.2361	1.23332	0.616659	−	0.787230i	$-0.288486\pi$
$83$	11.2361	1.23332	0.616659	+	0.787230i	$0.288486\pi$
$84$	0	0
$85$	−2.61803	−0.283966
$86$	0	0
$87$	3.23607	0.346943
$88$	0	0
$89$	−1.70820	−0.181069	−0.0905346	−	0.995893i	$-0.528858\pi$
$89$	−1.70820	−0.181069	−0.0905346	+	0.995893i	$0.528858\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.7082	1.52517
$94$	0	0
$95$	16.9443	1.73845
$96$	0	0
$97$	14.0344	1.42498	0.712491	−	0.701681i	$-0.247567\pi$
$97$	14.0344	1.42498	0.712491	+	0.701681i	$0.247567\pi$
$98$	0	0
$99$	−1.70820	−0.171681
$100$	0	0
$101$	16.9443	1.68602	0.843009	−	0.537899i	$-0.180782\pi$
$101$	16.9443	1.68602	0.843009	+	0.537899i	$0.180782\pi$
$102$	0	0
$103$	13.7082	1.35071	0.675355	−	0.737493i	$-0.263991\pi$
$103$	13.7082	1.35071	0.675355	+	0.737493i	$0.263991\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4721	−1.59242	−0.796211	−	0.605019i	$-0.793165\pi$
$107$	−16.4721	−1.59242	−0.796211	+	0.605019i	$0.793165\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	6.47214	0.614308
$112$	0	0
$113$	−5.70820	−0.536983	−0.268491	−	0.963282i	$-0.586525\pi$
$113$	−5.70820	−0.536983	−0.268491	+	0.963282i	$0.586525\pi$
$114$	0	0
$115$	−12.4721	−1.16303
$116$	0	0
$117$	1.70820	0.157924
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	16.3262	1.47209
$124$	0	0
$125$	−8.23607	−0.736656
$126$	0	0
$127$	4.90983	0.435677	0.217838	−	0.975985i	$-0.430099\pi$
$127$	4.90983	0.435677	0.217838	+	0.975985i	$0.430099\pi$
$128$	0	0
$129$	−16.3262	−1.43745
$130$	0	0
$131$	1.52786	0.133490	0.0667451	−	0.997770i	$-0.478739\pi$
$131$	1.52786	0.133490	0.0667451	+	0.997770i	$0.478739\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−14.3262	−1.23301
$136$	0	0
$137$	−16.0902	−1.37468	−0.687338	−	0.726338i	$-0.741221\pi$
$137$	−16.0902	−1.37468	−0.687338	+	0.726338i	$0.741221\pi$
$138$	0	0
$139$	−12.7984	−1.08554	−0.542772	−	0.839880i	$-0.682625\pi$
$139$	−12.7984	−1.08554	−0.542772	+	0.839880i	$0.682625\pi$
$140$	0	0
$141$	18.1803	1.53106
$142$	0	0
$143$	−20.0000	−1.67248
$144$	0	0
$145$	5.23607	0.434832
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.85410	0.725356	0.362678	−	0.931914i	$-0.381862\pi$
$149$	8.85410	0.725356	0.362678	+	0.931914i	$0.381862\pi$
$150$	0	0
$151$	1.32624	0.107928	0.0539639	−	0.998543i	$-0.482814\pi$
$151$	1.32624	0.107928	0.0539639	+	0.998543i	$0.482814\pi$
$152$	0	0
$153$	0.381966	0.0308801
$154$	0	0
$155$	23.7984	1.91153
$156$	0	0
$157$	−14.4721	−1.15500	−0.577501	−	0.816390i	$-0.695972\pi$
$157$	−14.4721	−1.15500	−0.577501	+	0.816390i	$0.695972\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.9443	−1.01387	−0.506937	−	0.861983i	$-0.669222\pi$
$163$	−12.9443	−1.01387	−0.506937	+	0.861983i	$0.669222\pi$
$164$	0	0
$165$	18.9443	1.47481
$166$	0	0
$167$	−7.32624	−0.566921	−0.283461	−	0.958984i	$-0.591483\pi$
$167$	−7.32624	−0.566921	−0.283461	+	0.958984i	$0.591483\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−2.47214	−0.189049
$172$	0	0
$173$	22.8541	1.73757	0.868783	−	0.495194i	$-0.164903\pi$
$173$	22.8541	1.73757	0.868783	+	0.495194i	$0.164903\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.47214	0.486476
$178$	0	0
$179$	−18.2705	−1.36560	−0.682801	−	0.730604i	$-0.739238\pi$
$179$	−18.2705	−1.36560	−0.682801	+	0.730604i	$0.739238\pi$
$180$	0	0
$181$	19.8885	1.47830	0.739152	−	0.673539i	$-0.235227\pi$
$181$	19.8885	1.47830	0.739152	+	0.673539i	$0.235227\pi$
$182$	0	0
$183$	−12.3262	−0.911182
$184$	0	0
$185$	10.4721	0.769927
$186$	0	0
$187$	−4.47214	−0.327035
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.32624	−0.602465	−0.301233	−	0.953551i	$-0.597398\pi$
$191$	−8.32624	−0.602465	−0.301233	+	0.953551i	$0.597398\pi$
$192$	0	0
$193$	−3.52786	−0.253941	−0.126971	−	0.991906i	$-0.540525\pi$
$193$	−3.52786	−0.253941	−0.126971	+	0.991906i	$0.540525\pi$
$194$	0	0
$195$	−18.9443	−1.35663
$196$	0	0
$197$	17.5279	1.24881	0.624404	−	0.781101i	$-0.285342\pi$
$197$	17.5279	1.24881	0.624404	+	0.781101i	$0.285342\pi$
$198$	0	0
$199$	−7.90983	−0.560713	−0.280356	−	0.959896i	$-0.590453\pi$
$199$	−7.90983	−0.560713	−0.280356	+	0.959896i	$0.590453\pi$
$200$	0	0
$201$	17.9443	1.26569
$202$	0	0
$203$	0	0
$204$	0	0
$205$	26.4164	1.84500
$206$	0	0
$207$	1.81966	0.126475
$208$	0	0
$209$	28.9443	2.00212
$210$	0	0
$211$	25.1246	1.72965	0.864825	−	0.502074i	$-0.167429\pi$
$211$	25.1246	1.72965	0.864825	+	0.502074i	$0.167429\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−26.4164	−1.80158
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.8541	−0.936173
$220$	0	0
$221$	4.47214	0.300828
$222$	0	0
$223$	12.9443	0.866813	0.433406	−	0.901199i	$-0.357312\pi$
$223$	12.9443	0.866813	0.433406	+	0.901199i	$0.357312\pi$
$224$	0	0
$225$	−0.708204	−0.0472136
$226$	0	0
$227$	0.145898	0.00968359	0.00484180	−	0.999988i	$-0.498459\pi$
$227$	0.145898	0.00968359	0.00484180	+	0.999988i	$0.498459\pi$
$228$	0	0
$229$	−19.5279	−1.29044	−0.645219	−	0.763998i	$-0.723234\pi$
$229$	−19.5279	−1.29044	−0.645219	+	0.763998i	$0.723234\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	29.4164	1.91891
$236$	0	0
$237$	13.2361	0.859775
$238$	0	0
$239$	9.32624	0.603264	0.301632	−	0.953424i	$-0.402469\pi$
$239$	9.32624	0.603264	0.301632	+	0.953424i	$0.402469\pi$
$240$	0	0
$241$	−12.7984	−0.824416	−0.412208	−	0.911090i	$-0.635242\pi$
$241$	−12.7984	−0.824416	−0.412208	+	0.911090i	$0.635242\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−28.9443	−1.84168
$248$	0	0
$249$	18.1803	1.15213
$250$	0	0
$251$	−13.8885	−0.876637	−0.438319	−	0.898820i	$-0.644426\pi$
$251$	−13.8885	−0.876637	−0.438319	+	0.898820i	$0.644426\pi$
$252$	0	0
$253$	−21.3050	−1.33943
$254$	0	0
$255$	−4.23607	−0.265273
$256$	0	0
$257$	−17.5967	−1.09765	−0.548827	−	0.835936i	$-0.684926\pi$
$257$	−17.5967	−1.09765	−0.548827	+	0.835936i	$0.684926\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.763932	−0.0472862
$262$	0	0
$263$	−26.8328	−1.65458	−0.827291	−	0.561773i	$-0.810119\pi$
$263$	−26.8328	−1.65458	−0.827291	+	0.561773i	$0.810119\pi$
$264$	0	0
$265$	−8.09017	−0.496975
$266$	0	0
$267$	−2.76393	−0.169150
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	0.763932	0.0464056	0.0232028	−	0.999731i	$-0.492614\pi$
$271$	0.763932	0.0464056	0.0232028	+	0.999731i	$0.492614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.29180	0.500014
$276$	0	0
$277$	−18.6525	−1.12072	−0.560359	−	0.828250i	$-0.689337\pi$
$277$	−18.6525	−1.12072	−0.560359	+	0.828250i	$0.689337\pi$
$278$	0	0
$279$	−3.47214	−0.207871
$280$	0	0
$281$	−1.85410	−0.110606	−0.0553032	−	0.998470i	$-0.517613\pi$
$281$	−1.85410	−0.110606	−0.0553032	+	0.998470i	$0.517613\pi$
$282$	0	0
$283$	13.7426	0.816915	0.408458	−	0.912777i	$-0.366067\pi$
$283$	13.7426	0.816915	0.408458	+	0.912777i	$0.366067\pi$
$284$	0	0
$285$	27.4164	1.62401
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	22.7082	1.33118
$292$	0	0
$293$	−14.1803	−0.828424	−0.414212	−	0.910180i	$-0.635943\pi$
$293$	−14.1803	−0.828424	−0.414212	+	0.910180i	$0.635943\pi$
$294$	0	0
$295$	10.4721	0.609711
$296$	0	0
$297$	−24.4721	−1.42002
$298$	0	0
$299$	21.3050	1.23210
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.4164	1.57503
$304$	0	0
$305$	−19.9443	−1.14201
$306$	0	0
$307$	14.7639	0.842622	0.421311	−	0.906916i	$-0.361570\pi$
$307$	14.7639	0.842622	0.421311	+	0.906916i	$0.361570\pi$
$308$	0	0
$309$	22.1803	1.26180
$310$	0	0
$311$	−32.5066	−1.84328	−0.921639	−	0.388047i	$-0.873150\pi$
$311$	−32.5066	−1.84328	−0.921639	+	0.388047i	$0.873150\pi$
$312$	0	0
$313$	−3.27051	−0.184860	−0.0924301	−	0.995719i	$-0.529463\pi$
$313$	−3.27051	−0.184860	−0.0924301	+	0.995719i	$0.529463\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.1246	1.74813	0.874066	−	0.485807i	$-0.161474\pi$
$317$	31.1246	1.74813	0.874066	+	0.485807i	$0.161474\pi$
$318$	0	0
$319$	8.94427	0.500783
$320$	0	0
$321$	−26.6525	−1.48760
$322$	0	0
$323$	−6.47214	−0.360119
$324$	0	0
$325$	−8.29180	−0.459946
$326$	0	0
$327$	16.1803	0.894775
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6.14590	−0.337809	−0.168905	−	0.985632i	$-0.554023\pi$
$331$	−6.14590	−0.337809	−0.168905	+	0.985632i	$0.554023\pi$
$332$	0	0
$333$	−1.52786	−0.0837264
$334$	0	0
$335$	29.0344	1.58632
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	−9.23607	−0.501634
$340$	0	0
$341$	40.6525	2.20145
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−20.1803	−1.08647
$346$	0	0
$347$	−26.3607	−1.41512	−0.707558	−	0.706656i	$-0.750203\pi$
$347$	−26.3607	−1.41512	−0.707558	+	0.706656i	$0.750203\pi$
$348$	0	0
$349$	18.1803	0.973171	0.486586	−	0.873633i	$-0.338242\pi$
$349$	18.1803	0.973171	0.486586	+	0.873633i	$0.338242\pi$
$350$	0	0
$351$	24.4721	1.30623
$352$	0	0
$353$	6.58359	0.350409	0.175205	−	0.984532i	$-0.443941\pi$
$353$	6.58359	0.350409	0.175205	+	0.984532i	$0.443941\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.20163	−0.274531	−0.137266	−	0.990534i	$-0.543831\pi$
$359$	−5.20163	−0.274531	−0.137266	+	0.990534i	$0.543831\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	14.5623	0.764323
$364$	0	0
$365$	−22.4164	−1.17333
$366$	0	0
$367$	−23.3262	−1.21762	−0.608810	−	0.793316i	$-0.708353\pi$
$367$	−23.3262	−1.21762	−0.608810	+	0.793316i	$0.708353\pi$
$368$	0	0
$369$	−3.85410	−0.200637
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.9098	−0.823780	−0.411890	−	0.911234i	$-0.635131\pi$
$373$	−15.9098	−0.823780	−0.411890	+	0.911234i	$0.635131\pi$
$374$	0	0
$375$	−13.3262	−0.688164
$376$	0	0
$377$	−8.94427	−0.460653
$378$	0	0
$379$	−31.3050	−1.60803	−0.804014	−	0.594611i	$-0.797306\pi$
$379$	−31.3050	−1.60803	−0.804014	+	0.594611i	$0.797306\pi$
$380$	0	0
$381$	7.94427	0.406997
$382$	0	0
$383$	−14.3607	−0.733796	−0.366898	−	0.930261i	$-0.619580\pi$
$383$	−14.3607	−0.733796	−0.366898	+	0.930261i	$0.619580\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.85410	0.195915
$388$	0	0
$389$	0.326238	0.0165409	0.00827046	−	0.999966i	$-0.497367\pi$
$389$	0.326238	0.0165409	0.00827046	+	0.999966i	$0.497367\pi$
$390$	0	0
$391$	4.76393	0.240922
$392$	0	0
$393$	2.47214	0.124703
$394$	0	0
$395$	21.4164	1.07758
$396$	0	0
$397$	−5.74265	−0.288215	−0.144108	−	0.989562i	$-0.546031\pi$
$397$	−5.74265	−0.288215	−0.144108	+	0.989562i	$0.546031\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.05573	0.0527205	0.0263603	−	0.999653i	$-0.491608\pi$
$401$	1.05573	0.0527205	0.0263603	+	0.999653i	$0.491608\pi$
$402$	0	0
$403$	−40.6525	−2.02504
$404$	0	0
$405$	−20.1803	−1.00277
$406$	0	0
$407$	17.8885	0.886702
$408$	0	0
$409$	38.0689	1.88239	0.941193	−	0.337871i	$-0.109707\pi$
$409$	38.0689	1.88239	0.941193	+	0.337871i	$0.109707\pi$
$410$	0	0
$411$	−26.0344	−1.28418
$412$	0	0
$413$	0	0
$414$	0	0
$415$	29.4164	1.44399
$416$	0	0
$417$	−20.7082	−1.01409
$418$	0	0
$419$	12.2148	0.596731	0.298366	−	0.954452i	$-0.403559\pi$
$419$	12.2148	0.596731	0.298366	+	0.954452i	$0.403559\pi$
$420$	0	0
$421$	29.0344	1.41505	0.707526	−	0.706687i	$-0.249811\pi$
$421$	29.0344	1.41505	0.707526	+	0.706687i	$0.249811\pi$
$422$	0	0
$423$	−4.29180	−0.208674
$424$	0	0
$425$	−1.85410	−0.0899372
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−32.3607	−1.56239
$430$	0	0
$431$	0.652476	0.0314287	0.0157143	−	0.999877i	$-0.494998\pi$
$431$	0.652476	0.0314287	0.0157143	+	0.999877i	$0.494998\pi$
$432$	0	0
$433$	4.29180	0.206251	0.103125	−	0.994668i	$-0.467116\pi$
$433$	4.29180	0.206251	0.103125	+	0.994668i	$0.467116\pi$
$434$	0	0
$435$	8.47214	0.406208
$436$	0	0
$437$	−30.8328	−1.47493
$438$	0	0
$439$	−6.56231	−0.313202	−0.156601	−	0.987662i	$-0.550054\pi$
$439$	−6.56231	−0.313202	−0.156601	+	0.987662i	$0.550054\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.5279	−1.40291	−0.701456	−	0.712713i	$-0.747466\pi$
$443$	−29.5279	−1.40291	−0.701456	+	0.712713i	$0.747466\pi$
$444$	0	0
$445$	−4.47214	−0.212000
$446$	0	0
$447$	14.3262	0.677608
$448$	0	0
$449$	−18.4721	−0.871754	−0.435877	−	0.900006i	$-0.643562\pi$
$449$	−18.4721	−0.871754	−0.435877	+	0.900006i	$0.643562\pi$
$450$	0	0
$451$	45.1246	2.12483
$452$	0	0
$453$	2.14590	0.100823
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.3262	−1.32505	−0.662523	−	0.749042i	$-0.730514\pi$
$457$	−28.3262	−1.32505	−0.662523	+	0.749042i	$0.730514\pi$
$458$	0	0
$459$	5.47214	0.255417
$460$	0	0
$461$	−26.1803	−1.21934	−0.609670	−	0.792655i	$-0.708698\pi$
$461$	−26.1803	−1.21934	−0.609670	+	0.792655i	$0.708698\pi$
$462$	0	0
$463$	−11.9787	−0.556698	−0.278349	−	0.960480i	$-0.589787\pi$
$463$	−11.9787	−0.556698	−0.278349	+	0.960480i	$0.589787\pi$
$464$	0	0
$465$	38.5066	1.78570
$466$	0	0
$467$	−22.1803	−1.02638	−0.513192	−	0.858274i	$-0.671537\pi$
$467$	−22.1803	−1.02638	−0.513192	+	0.858274i	$0.671537\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−23.4164	−1.07897
$472$	0	0
$473$	−45.1246	−2.07483
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	1.18034	0.0540441
$478$	0	0
$479$	−23.1459	−1.05756	−0.528782	−	0.848758i	$-0.677351\pi$
$479$	−23.1459	−1.05756	−0.528782	+	0.848758i	$0.677351\pi$
$480$	0	0
$481$	−17.8885	−0.815647
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.7426	1.66840
$486$	0	0
$487$	−5.23607	−0.237269	−0.118634	−	0.992938i	$-0.537852\pi$
$487$	−5.23607	−0.237269	−0.118634	+	0.992938i	$0.537852\pi$
$488$	0	0
$489$	−20.9443	−0.947133
$490$	0	0
$491$	10.5066	0.474155	0.237078	−	0.971491i	$-0.423810\pi$
$491$	10.5066	0.474155	0.237078	+	0.971491i	$0.423810\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	−4.47214	−0.201008
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.1803	0.455735	0.227867	−	0.973692i	$-0.426825\pi$
$499$	10.1803	0.455735	0.227867	+	0.973692i	$0.426825\pi$
$500$	0	0
$501$	−11.8541	−0.529602
$502$	0	0
$503$	−21.7426	−0.969457	−0.484728	−	0.874665i	$-0.661082\pi$
$503$	−21.7426	−0.969457	−0.484728	+	0.874665i	$0.661082\pi$
$504$	0	0
$505$	44.3607	1.97402
$506$	0	0
$507$	11.3262	0.503016
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−35.4164	−1.56367
$514$	0	0
$515$	35.8885	1.58144
$516$	0	0
$517$	50.2492	2.20996
$518$	0	0
$519$	36.9787	1.62319
$520$	0	0
$521$	18.8541	0.826013	0.413007	−	0.910728i	$-0.364479\pi$
$521$	18.8541	0.826013	0.413007	+	0.910728i	$0.364479\pi$
$522$	0	0
$523$	−17.7082	−0.774326	−0.387163	−	0.922011i	$-0.626545\pi$
$523$	−17.7082	−0.774326	−0.387163	+	0.922011i	$0.626545\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.09017	−0.395974
$528$	0	0
$529$	−0.304952	−0.0132588
$530$	0	0
$531$	−1.52786	−0.0663037
$532$	0	0
$533$	−45.1246	−1.95456
$534$	0	0
$535$	−43.1246	−1.86444
$536$	0	0
$537$	−29.5623	−1.27571
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.2918	1.30235	0.651173	−	0.758929i	$-0.274277\pi$
$541$	30.2918	1.30235	0.651173	+	0.758929i	$0.274277\pi$
$542$	0	0
$543$	32.1803	1.38099
$544$	0	0
$545$	26.1803	1.12144
$546$	0	0
$547$	38.8328	1.66037	0.830186	−	0.557487i	$-0.188234\pi$
$547$	38.8328	1.66037	0.830186	+	0.557487i	$0.188234\pi$
$548$	0	0
$549$	2.90983	0.124189
$550$	0	0
$551$	12.9443	0.551445
$552$	0	0
$553$	0	0
$554$	0	0
$555$	16.9443	0.719244
$556$	0	0
$557$	17.4164	0.737957	0.368978	−	0.929438i	$-0.379708\pi$
$557$	17.4164	0.737957	0.368978	+	0.929438i	$0.379708\pi$
$558$	0	0
$559$	45.1246	1.90857
$560$	0	0
$561$	−7.23607	−0.305507
$562$	0	0
$563$	−6.36068	−0.268071	−0.134035	−	0.990977i	$-0.542794\pi$
$563$	−6.36068	−0.268071	−0.134035	+	0.990977i	$0.542794\pi$
$564$	0	0
$565$	−14.9443	−0.628710
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.9230	−1.12867	−0.564335	−	0.825546i	$-0.690867\pi$
$569$	−26.9230	−1.12867	−0.564335	+	0.825546i	$0.690867\pi$
$570$	0	0
$571$	−10.4721	−0.438245	−0.219123	−	0.975697i	$-0.570319\pi$
$571$	−10.4721	−0.438245	−0.219123	+	0.975697i	$0.570319\pi$
$572$	0	0
$573$	−13.4721	−0.562807
$574$	0	0
$575$	−8.83282	−0.368354
$576$	0	0
$577$	3.05573	0.127212	0.0636058	−	0.997975i	$-0.479740\pi$
$577$	3.05573	0.127212	0.0636058	+	0.997975i	$0.479740\pi$
$578$	0	0
$579$	−5.70820	−0.237225
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.8197	−0.572352
$584$	0	0
$585$	4.47214	0.184900
$586$	0	0
$587$	−11.7082	−0.483249	−0.241625	−	0.970370i	$-0.577680\pi$
$587$	−11.7082	−0.483249	−0.241625	+	0.970370i	$0.577680\pi$
$588$	0	0
$589$	58.8328	2.42416
$590$	0	0
$591$	28.3607	1.16660
$592$	0	0
$593$	4.58359	0.188226	0.0941128	−	0.995562i	$-0.469999\pi$
$593$	4.58359	0.188226	0.0941128	+	0.995562i	$0.469999\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.7984	−0.523803
$598$	0	0
$599$	35.9787	1.47005	0.735025	−	0.678040i	$-0.237170\pi$
$599$	35.9787	1.47005	0.735025	+	0.678040i	$0.237170\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−4.23607	−0.172506
$604$	0	0
$605$	23.5623	0.957944
$606$	0	0
$607$	2.32624	0.0944191	0.0472095	−	0.998885i	$-0.484967\pi$
$607$	2.32624	0.0944191	0.0472095	+	0.998885i	$0.484967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−50.2492	−2.03287
$612$	0	0
$613$	−32.0344	−1.29386	−0.646929	−	0.762550i	$-0.723947\pi$
$613$	−32.0344	−1.29386	−0.646929	+	0.762550i	$0.723947\pi$
$614$	0	0
$615$	42.7426	1.72355
$616$	0	0
$617$	12.8328	0.516630	0.258315	−	0.966061i	$-0.416833\pi$
$617$	12.8328	0.516630	0.258315	+	0.966061i	$0.416833\pi$
$618$	0	0
$619$	−30.4721	−1.22478	−0.612389	−	0.790556i	$-0.709791\pi$
$619$	−30.4721	−1.22478	−0.612389	+	0.790556i	$0.709791\pi$
$620$	0	0
$621$	26.0689	1.04611
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	46.8328	1.87032
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	15.0902	0.600730	0.300365	−	0.953824i	$-0.402891\pi$
$631$	15.0902	0.600730	0.300365	+	0.953824i	$0.402891\pi$
$632$	0	0
$633$	40.6525	1.61579
$634$	0	0
$635$	12.8541	0.510099
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.7639	0.425150	0.212575	−	0.977145i	$-0.431815\pi$
$641$	10.7639	0.425150	0.212575	+	0.977145i	$0.431815\pi$
$642$	0	0
$643$	−18.5623	−0.732026	−0.366013	−	0.930610i	$-0.619277\pi$
$643$	−18.5623	−0.732026	−0.366013	+	0.930610i	$0.619277\pi$
$644$	0	0
$645$	−42.7426	−1.68299
$646$	0	0
$647$	22.5836	0.887853	0.443926	−	0.896063i	$-0.353585\pi$
$647$	22.5836	0.887853	0.443926	+	0.896063i	$0.353585\pi$
$648$	0	0
$649$	17.8885	0.702187
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.8885	−0.778299	−0.389149	−	0.921175i	$-0.627231\pi$
$653$	−19.8885	−0.778299	−0.389149	+	0.921175i	$0.627231\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	3.27051	0.127595
$658$	0	0
$659$	−0.326238	−0.0127084	−0.00635421	−	0.999980i	$-0.502023\pi$
$659$	−0.326238	−0.0127084	−0.00635421	+	0.999980i	$0.502023\pi$
$660$	0	0
$661$	−31.3050	−1.21762	−0.608811	−	0.793315i	$-0.708353\pi$
$661$	−31.3050	−1.21762	−0.608811	+	0.793315i	$0.708353\pi$
$662$	0	0
$663$	7.23607	0.281026
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.52786	−0.368920
$668$	0	0
$669$	20.9443	0.809752
$670$	0	0
$671$	−34.0689	−1.31521
$672$	0	0
$673$	31.7082	1.22226	0.611131	−	0.791530i	$-0.290715\pi$
$673$	31.7082	1.22226	0.611131	+	0.791530i	$0.290715\pi$
$674$	0	0
$675$	−10.1459	−0.390516
$676$	0	0
$677$	1.63932	0.0630042	0.0315021	−	0.999504i	$-0.489971\pi$
$677$	1.63932	0.0630042	0.0315021	+	0.999504i	$0.489971\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.236068	0.00904614
$682$	0	0
$683$	−16.9443	−0.648355	−0.324177	−	0.945996i	$-0.605087\pi$
$683$	−16.9443	−0.648355	−0.324177	+	0.945996i	$0.605087\pi$
$684$	0	0
$685$	−42.1246	−1.60950
$686$	0	0
$687$	−31.5967	−1.20549
$688$	0	0
$689$	13.8197	0.526487
$690$	0	0
$691$	−4.85410	−0.184659	−0.0923294	−	0.995729i	$-0.529431\pi$
$691$	−4.85410	−0.184659	−0.0923294	+	0.995729i	$0.529431\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−33.5066	−1.27098
$696$	0	0
$697$	−10.0902	−0.382192
$698$	0	0
$699$	9.70820	0.367198
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	25.8885	0.976405
$704$	0	0
$705$	47.5967	1.79260
$706$	0	0
$707$	0	0
$708$	0	0
$709$	33.4164	1.25498	0.627490	−	0.778625i	$-0.284082\pi$
$709$	33.4164	1.25498	0.627490	+	0.778625i	$0.284082\pi$
$710$	0	0
$711$	−3.12461	−0.117182
$712$	0	0
$713$	−43.3050	−1.62178
$714$	0	0
$715$	−52.3607	−1.95818
$716$	0	0
$717$	15.0902	0.563553
$718$	0	0
$719$	−11.3262	−0.422397	−0.211199	−	0.977443i	$-0.567737\pi$
$719$	−11.3262	−0.422397	−0.211199	+	0.977443i	$0.567737\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−20.7082	−0.770146
$724$	0	0
$725$	3.70820	0.137719
$726$	0	0
$727$	13.8197	0.512543	0.256271	−	0.966605i	$-0.417506\pi$
$727$	13.8197	0.512543	0.256271	+	0.966605i	$0.417506\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	10.0902	0.373198
$732$	0	0
$733$	−2.40325	−0.0887661	−0.0443831	−	0.999015i	$-0.514132\pi$
$733$	−2.40325	−0.0887661	−0.0443831	+	0.999015i	$0.514132\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49.5967	1.82692
$738$	0	0
$739$	16.2016	0.595986	0.297993	−	0.954568i	$-0.403683\pi$
$739$	16.2016	0.595986	0.297993	+	0.954568i	$0.403683\pi$
$740$	0	0
$741$	−46.8328	−1.72045
$742$	0	0
$743$	6.76393	0.248145	0.124072	−	0.992273i	$-0.460405\pi$
$743$	6.76393	0.248145	0.124072	+	0.992273i	$0.460405\pi$
$744$	0	0
$745$	23.1803	0.849262
$746$	0	0
$747$	−4.29180	−0.157029
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	−22.4721	−0.818930
$754$	0	0
$755$	3.47214	0.126364
$756$	0	0
$757$	4.67376	0.169871	0.0849354	−	0.996386i	$-0.472932\pi$
$757$	4.67376	0.169871	0.0849354	+	0.996386i	$0.472932\pi$
$758$	0	0
$759$	−34.4721	−1.25126
$760$	0	0
$761$	53.9574	1.95596	0.977978	−	0.208710i	$-0.0669264\pi$
$761$	53.9574	1.95596	0.977978	+	0.208710i	$0.0669264\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−17.8885	−0.645918
$768$	0	0
$769$	−54.2492	−1.95628	−0.978139	−	0.207954i	$-0.933319\pi$
$769$	−54.2492	−1.95628	−0.978139	+	0.207954i	$0.933319\pi$
$770$	0	0
$771$	−28.4721	−1.02540
$772$	0	0
$773$	−27.8197	−1.00060	−0.500302	−	0.865851i	$-0.666778\pi$
$773$	−27.8197	−1.00060	−0.500302	+	0.865851i	$0.666778\pi$
$774$	0	0
$775$	16.8541	0.605417
$776$	0	0
$777$	0	0
$778$	0	0
$779$	65.3050	2.33979
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−10.9443	−0.391116
$784$	0	0
$785$	−37.8885	−1.35230
$786$	0	0
$787$	22.4721	0.801045	0.400523	−	0.916287i	$-0.368829\pi$
$787$	22.4721	0.801045	0.400523	+	0.916287i	$0.368829\pi$
$788$	0	0
$789$	−43.4164	−1.54567
$790$	0	0
$791$	0	0
$792$	0	0
$793$	34.0689	1.20982
$794$	0	0
$795$	−13.0902	−0.464260
$796$	0	0
$797$	9.12461	0.323210	0.161605	−	0.986855i	$-0.448333\pi$
$797$	9.12461	0.323210	0.161605	+	0.986855i	$0.448333\pi$
$798$	0	0
$799$	−11.2361	−0.397504
$800$	0	0
$801$	0.652476	0.0230541
$802$	0	0
$803$	−38.2918	−1.35129
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.70820	−0.341745
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	29.7984	1.04636	0.523181	−	0.852221i	$-0.324745\pi$
$811$	29.7984	1.04636	0.523181	+	0.852221i	$0.324745\pi$
$812$	0	0
$813$	1.23607	0.0433508
$814$	0	0
$815$	−33.8885	−1.18706
$816$	0	0
$817$	−65.3050	−2.28473
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.4164	−0.956839	−0.478420	−	0.878131i	$-0.658790\pi$
$821$	−27.4164	−0.956839	−0.478420	+	0.878131i	$0.658790\pi$
$822$	0	0
$823$	−2.40325	−0.0837721	−0.0418861	−	0.999122i	$-0.513337\pi$
$823$	−2.40325	−0.0837721	−0.0418861	+	0.999122i	$0.513337\pi$
$824$	0	0
$825$	13.4164	0.467099
$826$	0	0
$827$	−6.76393	−0.235205	−0.117602	−	0.993061i	$-0.537521\pi$
$827$	−6.76393	−0.235205	−0.117602	+	0.993061i	$0.537521\pi$
$828$	0	0
$829$	−45.7771	−1.58990	−0.794952	−	0.606672i	$-0.792504\pi$
$829$	−45.7771	−1.58990	−0.794952	+	0.606672i	$0.792504\pi$
$830$	0	0
$831$	−30.1803	−1.04694
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19.1803	−0.663763
$836$	0	0
$837$	−49.7426	−1.71936
$838$	0	0
$839$	18.4721	0.637729	0.318864	−	0.947800i	$-0.396699\pi$
$839$	18.4721	0.637729	0.318864	+	0.947800i	$0.396699\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−3.00000	−0.103325
$844$	0	0
$845$	18.3262	0.630442
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.2361	0.763140
$850$	0	0
$851$	−19.0557	−0.653222
$852$	0	0
$853$	37.7771	1.29346	0.646731	−	0.762718i	$-0.276135\pi$
$853$	37.7771	1.29346	0.646731	+	0.762718i	$0.276135\pi$
$854$	0	0
$855$	−6.47214	−0.221342
$856$	0	0
$857$	3.27051	0.111718	0.0558592	−	0.998439i	$-0.482210\pi$
$857$	3.27051	0.111718	0.0558592	+	0.998439i	$0.482210\pi$
$858$	0	0
$859$	22.2918	0.760586	0.380293	−	0.924866i	$-0.375823\pi$
$859$	22.2918	0.760586	0.380293	+	0.924866i	$0.375823\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.2705	0.587895	0.293947	−	0.955822i	$-0.405031\pi$
$863$	17.2705	0.587895	0.293947	+	0.955822i	$0.405031\pi$
$864$	0	0
$865$	59.8328	2.03438
$866$	0	0
$867$	1.61803	0.0549513
$868$	0	0
$869$	36.5836	1.24101
$870$	0	0
$871$	−49.5967	−1.68052
$872$	0	0
$873$	−5.36068	−0.181432
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.5836	0.965199	0.482600	−	0.875841i	$-0.339693\pi$
$877$	28.5836	0.965199	0.482600	+	0.875841i	$0.339693\pi$
$878$	0	0
$879$	−22.9443	−0.773891
$880$	0	0
$881$	−56.5066	−1.90375	−0.951877	−	0.306479i	$-0.900849\pi$
$881$	−56.5066	−1.90375	−0.951877	+	0.306479i	$0.900849\pi$
$882$	0	0
$883$	41.6312	1.40100	0.700501	−	0.713652i	$-0.252960\pi$
$883$	41.6312	1.40100	0.700501	+	0.713652i	$0.252960\pi$
$884$	0	0
$885$	16.9443	0.569575
$886$	0	0
$887$	14.9787	0.502936	0.251468	−	0.967866i	$-0.419087\pi$
$887$	14.9787	0.502936	0.251468	+	0.967866i	$0.419087\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−34.4721	−1.15486
$892$	0	0
$893$	72.7214	2.43353
$894$	0	0
$895$	−47.8328	−1.59887
$896$	0	0
$897$	34.4721	1.15099
$898$	0	0
$899$	18.1803	0.606348
$900$	0	0
$901$	3.09017	0.102948
$902$	0	0
$903$	0	0
$904$	0	0
$905$	52.0689	1.73083
$906$	0	0
$907$	−39.3050	−1.30510	−0.652550	−	0.757746i	$-0.726301\pi$
$907$	−39.3050	−1.30510	−0.652550	+	0.757746i	$0.726301\pi$
$908$	0	0
$909$	−6.47214	−0.214667
$910$	0	0
$911$	−3.52786	−0.116883	−0.0584417	−	0.998291i	$-0.518613\pi$
$911$	−3.52786	−0.116883	−0.0584417	+	0.998291i	$0.518613\pi$
$912$	0	0
$913$	50.2492	1.66301
$914$	0	0
$915$	−32.2705	−1.06683
$916$	0	0
$917$	0	0
$918$	0	0
$919$	44.9230	1.48187	0.740936	−	0.671575i	$-0.234382\pi$
$919$	44.9230	1.48187	0.740936	+	0.671575i	$0.234382\pi$
$920$	0	0
$921$	23.8885	0.787154
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.41641	0.243850
$926$	0	0
$927$	−5.23607	−0.171975
$928$	0	0
$929$	38.1033	1.25013	0.625065	−	0.780573i	$-0.285073\pi$
$929$	38.1033	1.25013	0.625065	+	0.780573i	$0.285073\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−52.5967	−1.72194
$934$	0	0
$935$	−11.7082	−0.382899
$936$	0	0
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	−5.29180	−0.172691
$940$	0	0
$941$	20.7426	0.676191	0.338095	−	0.941112i	$-0.390217\pi$
$941$	20.7426	0.676191	0.338095	+	0.941112i	$0.390217\pi$
$942$	0	0
$943$	−48.0689	−1.56534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.8328	1.84682	0.923409	−	0.383817i	$-0.125391\pi$
$947$	56.8328	1.84682	0.923409	+	0.383817i	$0.125391\pi$
$948$	0	0
$949$	38.2918	1.24300
$950$	0	0
$951$	50.3607	1.63306
$952$	0	0
$953$	−17.5623	−0.568899	−0.284449	−	0.958691i	$-0.591811\pi$
$953$	−17.5623	−0.568899	−0.284449	+	0.958691i	$0.591811\pi$
$954$	0	0
$955$	−21.7984	−0.705379
$956$	0	0
$957$	14.4721	0.467818
$958$	0	0
$959$	0	0
$960$	0	0
$961$	51.6312	1.66552
$962$	0	0
$963$	6.29180	0.202750
$964$	0	0
$965$	−9.23607	−0.297320
$966$	0	0
$967$	−36.5623	−1.17576	−0.587882	−	0.808947i	$-0.700038\pi$
$967$	−36.5623	−1.17576	−0.587882	+	0.808947i	$0.700038\pi$
$968$	0	0
$969$	−10.4721	−0.336413
$970$	0	0
$971$	−17.4164	−0.558919	−0.279460	−	0.960157i	$-0.590155\pi$
$971$	−17.4164	−0.558919	−0.279460	+	0.960157i	$0.590155\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−13.4164	−0.429669
$976$	0	0
$977$	2.50658	0.0801925	0.0400963	−	0.999196i	$-0.487234\pi$
$977$	2.50658	0.0801925	0.0400963	+	0.999196i	$0.487234\pi$
$978$	0	0
$979$	−7.63932	−0.244154
$980$	0	0
$981$	−3.81966	−0.121952
$982$	0	0
$983$	28.5066	0.909219	0.454609	−	0.890691i	$-0.349779\pi$
$983$	28.5066	0.909219	0.454609	+	0.890691i	$0.349779\pi$
$984$	0	0
$985$	45.8885	1.46213
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.0689	1.52850
$990$	0	0
$991$	18.5836	0.590327	0.295164	−	0.955447i	$-0.404626\pi$
$991$	18.5836	0.590327	0.295164	+	0.955447i	$0.404626\pi$
$992$	0	0
$993$	−9.94427	−0.315572
$994$	0	0
$995$	−20.7082	−0.656494
$996$	0	0
$997$	35.4377	1.12232	0.561162	−	0.827706i	$-0.310355\pi$
$997$	35.4377	1.12232	0.561162	+	0.827706i	$0.310355\pi$
$998$	0	0
$999$	−21.8885	−0.692523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.h.1.2		2
7.6	odd	2		952.2.a.a.1.1	✓	2
21.20	even	2		8568.2.a.v.1.2		2
28.27	even	2		1904.2.a.i.1.2		2
56.13	odd	2		7616.2.a.w.1.2		2
56.27	even	2		7616.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.a.1.1	✓	2	7.6	odd	2
1904.2.a.i.1.2		2	28.27	even	2
6664.2.a.h.1.2		2	1.1	even	1	trivial
7616.2.a.r.1.1		2	56.27	even	2
7616.2.a.w.1.2		2	56.13	odd	2
8568.2.a.v.1.2		2	21.20	even	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:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} +2.61803 q^{5} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} +2.61803 q^{5} -0.381966 q^{9} +4.47214 q^{11} -4.47214 q^{13} +4.23607 q^{15} -1.00000 q^{17} +6.47214 q^{19} -4.76393 q^{23} +1.85410 q^{25} -5.47214 q^{27} +2.00000 q^{29} +9.09017 q^{31} +7.23607 q^{33} +4.00000 q^{37} -7.23607 q^{39} +10.0902 q^{41} -10.0902 q^{43} -1.00000 q^{45} +11.2361 q^{47} -1.61803 q^{51} -3.09017 q^{53} +11.7082 q^{55} +10.4721 q^{57} +4.00000 q^{59} -7.61803 q^{61} -11.7082 q^{65} +11.0902 q^{67} -7.70820 q^{69} -8.56231 q^{73} +3.00000 q^{75} +8.18034 q^{79} -7.70820 q^{81} +11.2361 q^{83} -2.61803 q^{85} +3.23607 q^{87} -1.70820 q^{89} +14.7082 q^{93} +16.9443 q^{95} +14.0344 q^{97} -1.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 3 * q^5 - 3 * q^9	\( 2 q + q^{3} + 3 q^{5} - 3 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{19} - 14 q^{23} - 3 q^{25} - 2 q^{27} + 4 q^{29} + 7 q^{31} + 10 q^{33} + 8 q^{37} - 10 q^{39} + 9 q^{41} - 9 q^{43} - 2 q^{45} + 18 q^{47} - q^{51} + 5 q^{53} + 10 q^{55} + 12 q^{57} + 8 q^{59} - 13 q^{61} - 10 q^{65} + 11 q^{67} - 2 q^{69} + 3 q^{73} + 6 q^{75} - 6 q^{79} - 2 q^{81} + 18 q^{83} - 3 q^{85} + 2 q^{87} + 10 q^{89} + 16 q^{93} + 16 q^{95} - q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q + q^3 + 3 * q^5 - 3 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^19 - 14 * q^23 - 3 * q^25 - 2 * q^27 + 4 * q^29 + 7 * q^31 + 10 * q^33 + 8 * q^37 - 10 * q^39 + 9 * q^41 - 9 * q^43 - 2 * q^45 + 18 * q^47 - q^51 + 5 * q^53 + 10 * q^55 + 12 * q^57 + 8 * q^59 - 13 * q^61 - 10 * q^65 + 11 * q^67 - 2 * q^69 + 3 * q^73 + 6 * q^75 - 6 * q^79 - 2 * q^81 + 18 * q^83 - 3 * q^85 + 2 * q^87 + 10 * q^89 + 16 * q^93 + 16 * q^95 - q^97 + 10 * q^99

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