LMFDB - Embedded newform 7728.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} -2.00000 q^{15} -4.47214 q^{17} +6.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.47214 q^{31} -2.00000 q^{35} -10.9443 q^{37} -4.47214 q^{39} -6.00000 q^{41} -12.9443 q^{43} +2.00000 q^{45} -6.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} +6.94427 q^{53} -6.47214 q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} +8.94427 q^{65} +12.9443 q^{67} -1.00000 q^{69} +12.9443 q^{71} -2.94427 q^{73} +1.00000 q^{75} -12.9443 q^{79} +1.00000 q^{81} +6.47214 q^{83} -8.94427 q^{85} +2.00000 q^{87} -17.4164 q^{89} -4.47214 q^{91} +6.47214 q^{93} +12.9443 q^{95} +8.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 + 4 * q^19 + 2 * q^21 + 2 * q^23 - 2 * q^25 - 2 * q^27 - 4 * q^29 - 4 * q^31 - 4 * q^35 - 4 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 - 4 * q^47 + 2 * q^49 - 4 * q^53 - 4 * q^57 - 8 * q^59 - 12 * q^61 - 2 * q^63 + 8 * q^67 - 2 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^87 - 8 * q^89 + 4 * q^93 + 8 * q^95 + 8 * q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	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
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−12.9443	−1.97398	−0.986991	−	0.160773i	$-0.948601\pi$
$43$	−12.9443	−1.97398	−0.986991	+	0.160773i	$0.948601\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	6.94427	0.953869	0.476935	−	0.878939i	$-0.341748\pi$
$53$	6.94427	0.953869	0.476935	+	0.878939i	$0.341748\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.47214	−0.857255
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	8.94427	1.10940
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.9443	1.53620	0.768101	−	0.640328i	$-0.221202\pi$
$71$	12.9443	1.53620	0.768101	+	0.640328i	$0.221202\pi$
$72$	0	0
$73$	−2.94427	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.47214	0.710409	0.355205	−	0.934789i	$-0.384411\pi$
$83$	6.47214	0.710409	0.355205	+	0.934789i	$0.384411\pi$
$84$	0	0
$85$	−8.94427	−0.970143
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−17.4164	−1.84614	−0.923068	−	0.384637i	$-0.874327\pi$
$89$	−17.4164	−1.84614	−0.923068	+	0.384637i	$0.874327\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	6.47214	0.671129
$94$	0	0
$95$	12.9443	1.32805
$96$	0	0
$97$	8.47214	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$97$	8.47214	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.4164	1.73300	0.866499	−	0.499179i	$-0.166365\pi$
$101$	17.4164	1.73300	0.866499	+	0.499179i	$0.166365\pi$
$102$	0	0
$103$	−12.9443	−1.27544	−0.637719	−	0.770270i	$-0.720122\pi$
$103$	−12.9443	−1.27544	−0.637719	+	0.770270i	$0.720122\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	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
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	10.9443	1.03878
$112$	0	0
$113$	−1.05573	−0.0993145	−0.0496573	−	0.998766i	$-0.515813\pi$
$113$	−1.05573	−0.0993145	−0.0496573	+	0.998766i	$0.515813\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	4.47214	0.413449
$118$	0	0
$119$	4.47214	0.409960
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−3.05573	−0.271152	−0.135576	−	0.990767i	$-0.543288\pi$
$127$	−3.05573	−0.271152	−0.135576	+	0.990767i	$0.543288\pi$
$128$	0	0
$129$	12.9443	1.13968
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−6.47214	−0.561205
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−18.9443	−1.55198	−0.775988	−	0.630748i	$-0.782748\pi$
$149$	−18.9443	−1.55198	−0.775988	+	0.630748i	$0.782748\pi$
$150$	0	0
$151$	12.9443	1.05339	0.526695	−	0.850054i	$-0.323431\pi$
$151$	12.9443	1.05339	0.526695	+	0.850054i	$0.323431\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	−12.9443	−1.03971
$156$	0	0
$157$	14.9443	1.19268	0.596341	−	0.802731i	$-0.296620\pi$
$157$	14.9443	1.19268	0.596341	+	0.802731i	$0.296620\pi$
$158$	0	0
$159$	−6.94427	−0.550717
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−8.94427	−0.700569	−0.350285	−	0.936643i	$-0.613915\pi$
$163$	−8.94427	−0.700569	−0.350285	+	0.936643i	$0.613915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.4164	−0.883428	−0.441714	−	0.897156i	$-0.645629\pi$
$167$	−11.4164	−0.883428	−0.441714	+	0.897156i	$0.645629\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	6.47214	0.494937
$172$	0	0
$173$	−8.47214	−0.644125	−0.322062	−	0.946718i	$-0.604376\pi$
$173$	−8.47214	−0.644125	−0.322062	+	0.946718i	$0.604376\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−21.8885	−1.60928
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0	0
$193$	−23.8885	−1.71954	−0.859768	−	0.510686i	$-0.829392\pi$
$193$	−23.8885	−1.71954	−0.859768	+	0.510686i	$0.829392\pi$
$194$	0	0
$195$	−8.94427	−0.640513
$196$	0	0
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	0	0
$199$	−20.9443	−1.48470	−0.742350	−	0.670012i	$-0.766289\pi$
$199$	−20.9443	−1.48470	−0.742350	+	0.670012i	$0.766289\pi$
$200$	0	0
$201$	−12.9443	−0.913019
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	0	0
$213$	−12.9443	−0.886927
$214$	0	0
$215$	−25.8885	−1.76558
$216$	0	0
$217$	6.47214	0.439357
$218$	0	0
$219$	2.94427	0.198955
$220$	0	0
$221$	−20.0000	−1.34535
$222$	0	0
$223$	−9.52786	−0.638033	−0.319016	−	0.947749i	$-0.603353\pi$
$223$	−9.52786	−0.638033	−0.319016	+	0.947749i	$0.603353\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−14.4721	−0.960549	−0.480275	−	0.877118i	$-0.659463\pi$
$227$	−14.4721	−0.960549	−0.480275	+	0.877118i	$0.659463\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−12.9443	−0.844391
$236$	0	0
$237$	12.9443	0.840821
$238$	0	0
$239$	−3.05573	−0.197659	−0.0988293	−	0.995104i	$-0.531510\pi$
$239$	−3.05573	−0.197659	−0.0988293	+	0.995104i	$0.531510\pi$
$240$	0	0
$241$	11.5279	0.742575	0.371288	−	0.928518i	$-0.378916\pi$
$241$	11.5279	0.742575	0.371288	+	0.928518i	$0.378916\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	28.9443	1.84168
$248$	0	0
$249$	−6.47214	−0.410155
$250$	0	0
$251$	14.4721	0.913473	0.456737	−	0.889602i	$-0.349018\pi$
$251$	14.4721	0.913473	0.456737	+	0.889602i	$0.349018\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.94427	0.560112
$256$	0	0
$257$	14.9443	0.932198	0.466099	−	0.884733i	$-0.345659\pi$
$257$	14.9443	0.932198	0.466099	+	0.884733i	$0.345659\pi$
$258$	0	0
$259$	10.9443	0.680044
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	13.8885	0.853166
$266$	0	0
$267$	17.4164	1.06587
$268$	0	0
$269$	7.52786	0.458982	0.229491	−	0.973311i	$-0.426294\pi$
$269$	7.52786	0.458982	0.229491	+	0.973311i	$0.426294\pi$
$270$	0	0
$271$	11.4164	0.693497	0.346749	−	0.937958i	$-0.387286\pi$
$271$	11.4164	0.693497	0.346749	+	0.937958i	$0.387286\pi$
$272$	0	0
$273$	4.47214	0.270666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.8885	1.43532	0.717662	−	0.696392i	$-0.245212\pi$
$277$	23.8885	1.43532	0.717662	+	0.696392i	$0.245212\pi$
$278$	0	0
$279$	−6.47214	−0.387477
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−9.52786	−0.566373	−0.283186	−	0.959065i	$-0.591391\pi$
$283$	−9.52786	−0.566373	−0.283186	+	0.959065i	$0.591391\pi$
$284$	0	0
$285$	−12.9443	−0.766752
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−8.47214	−0.496645
$292$	0	0
$293$	−7.88854	−0.460854	−0.230427	−	0.973090i	$-0.574012\pi$
$293$	−7.88854	−0.460854	−0.230427	+	0.973090i	$0.574012\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.47214	0.258630
$300$	0	0
$301$	12.9443	0.746095
$302$	0	0
$303$	−17.4164	−1.00055
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−7.05573	−0.402692	−0.201346	−	0.979520i	$-0.564532\pi$
$307$	−7.05573	−0.402692	−0.201346	+	0.979520i	$0.564532\pi$
$308$	0	0
$309$	12.9443	0.736374
$310$	0	0
$311$	−6.47214	−0.367001	−0.183501	−	0.983020i	$-0.558743\pi$
$311$	−6.47214	−0.367001	−0.183501	+	0.983020i	$0.558743\pi$
$312$	0	0
$313$	13.4164	0.758340	0.379170	−	0.925327i	$-0.376210\pi$
$313$	13.4164	0.758340	0.379170	+	0.925327i	$0.376210\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−30.9443	−1.73800	−0.869002	−	0.494809i	$-0.835238\pi$
$317$	−30.9443	−1.73800	−0.869002	+	0.494809i	$0.835238\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−28.9443	−1.61050
$324$	0	0
$325$	−4.47214	−0.248069
$326$	0	0
$327$	−14.9443	−0.826420
$328$	0	0
$329$	6.47214	0.356820
$330$	0	0
$331$	−21.8885	−1.20310	−0.601552	−	0.798834i	$-0.705451\pi$
$331$	−21.8885	−1.20310	−0.601552	+	0.798834i	$0.705451\pi$
$332$	0	0
$333$	−10.9443	−0.599742
$334$	0	0
$335$	25.8885	1.41444
$336$	0	0
$337$	14.9443	0.814066	0.407033	−	0.913413i	$-0.366563\pi$
$337$	14.9443	0.814066	0.407033	+	0.913413i	$0.366563\pi$
$338$	0	0
$339$	1.05573	0.0573393
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	−16.9443	−0.909616	−0.454808	−	0.890589i	$-0.650292\pi$
$347$	−16.9443	−0.909616	−0.454808	+	0.890589i	$0.650292\pi$
$348$	0	0
$349$	15.5279	0.831188	0.415594	−	0.909550i	$-0.363574\pi$
$349$	15.5279	0.831188	0.415594	+	0.909550i	$0.363574\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	25.8885	1.37402
$356$	0	0
$357$	−4.47214	−0.236691
$358$	0	0
$359$	−22.8328	−1.20507	−0.602535	−	0.798092i	$-0.705843\pi$
$359$	−22.8328	−1.20507	−0.602535	+	0.798092i	$0.705843\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−5.88854	−0.308220
$366$	0	0
$367$	22.8328	1.19186	0.595932	−	0.803035i	$-0.296783\pi$
$367$	22.8328	1.19186	0.595932	+	0.803035i	$0.296783\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−6.94427	−0.360529
$372$	0	0
$373$	−10.9443	−0.566673	−0.283336	−	0.959021i	$-0.591441\pi$
$373$	−10.9443	−0.566673	−0.283336	+	0.959021i	$0.591441\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−8.94427	−0.460653
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	3.05573	0.156550
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.9443	−0.657994
$388$	0	0
$389$	−1.05573	−0.0535275	−0.0267638	−	0.999642i	$-0.508520\pi$
$389$	−1.05573	−0.0535275	−0.0267638	+	0.999642i	$0.508520\pi$
$390$	0	0
$391$	−4.47214	−0.226166
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	−25.8885	−1.30259
$396$	0	0
$397$	−27.5279	−1.38158	−0.690792	−	0.723054i	$-0.742738\pi$
$397$	−27.5279	−1.38158	−0.690792	+	0.723054i	$0.742738\pi$
$398$	0	0
$399$	6.47214	0.324012
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	−28.9443	−1.44182
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	0	0
$408$	0	0
$409$	19.8885	0.983425	0.491713	−	0.870758i	$-0.336371\pi$
$409$	19.8885	0.983425	0.491713	+	0.870758i	$0.336371\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	12.9443	0.635409
$416$	0	0
$417$	−8.94427	−0.438003
$418$	0	0
$419$	−19.4164	−0.948554	−0.474277	−	0.880376i	$-0.657290\pi$
$419$	−19.4164	−0.948554	−0.474277	+	0.880376i	$0.657290\pi$
$420$	0	0
$421$	−20.8328	−1.01533	−0.507665	−	0.861555i	$-0.669491\pi$
$421$	−20.8328	−1.01533	−0.507665	+	0.861555i	$0.669491\pi$
$422$	0	0
$423$	−6.47214	−0.314686
$424$	0	0
$425$	4.47214	0.216930
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	0	0
$433$	−1.41641	−0.0680682	−0.0340341	−	0.999421i	$-0.510835\pi$
$433$	−1.41641	−0.0680682	−0.0340341	+	0.999421i	$0.510835\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	6.47214	0.309604
$438$	0	0
$439$	16.3607	0.780853	0.390426	−	0.920634i	$-0.372328\pi$
$439$	16.3607	0.780853	0.390426	+	0.920634i	$0.372328\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.8328	0.894774	0.447387	−	0.894340i	$-0.352355\pi$
$443$	18.8328	0.894774	0.447387	+	0.894340i	$0.352355\pi$
$444$	0	0
$445$	−34.8328	−1.65123
$446$	0	0
$447$	18.9443	0.896033
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−12.9443	−0.608175
$454$	0	0
$455$	−8.94427	−0.419314
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	33.4164	1.55636	0.778179	−	0.628043i	$-0.216144\pi$
$461$	33.4164	1.55636	0.778179	+	0.628043i	$0.216144\pi$
$462$	0	0
$463$	−41.8885	−1.94673	−0.973363	−	0.229270i	$-0.926366\pi$
$463$	−41.8885	−1.94673	−0.973363	+	0.229270i	$0.926366\pi$
$464$	0	0
$465$	12.9443	0.600276
$466$	0	0
$467$	29.3050	1.35607	0.678036	−	0.735029i	$-0.262831\pi$
$467$	29.3050	1.35607	0.678036	+	0.735029i	$0.262831\pi$
$468$	0	0
$469$	−12.9443	−0.597711
$470$	0	0
$471$	−14.9443	−0.688596
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.47214	−0.296962
$476$	0	0
$477$	6.94427	0.317956
$478$	0	0
$479$	−12.9443	−0.591439	−0.295719	−	0.955275i	$-0.595559\pi$
$479$	−12.9443	−0.591439	−0.295719	+	0.955275i	$0.595559\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	16.9443	0.769400
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	8.94427	0.404474
$490$	0	0
$491$	−23.0557	−1.04049	−0.520245	−	0.854017i	$-0.674159\pi$
$491$	−23.0557	−1.04049	−0.520245	+	0.854017i	$0.674159\pi$
$492$	0	0
$493$	8.94427	0.402830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.9443	−0.580630
$498$	0	0
$499$	−15.0557	−0.673987	−0.336993	−	0.941507i	$-0.609410\pi$
$499$	−15.0557	−0.673987	−0.336993	+	0.941507i	$0.609410\pi$
$500$	0	0
$501$	11.4164	0.510047
$502$	0	0
$503$	−25.8885	−1.15431	−0.577157	−	0.816634i	$-0.695838\pi$
$503$	−25.8885	−1.15431	−0.577157	+	0.816634i	$0.695838\pi$
$504$	0	0
$505$	34.8328	1.55004
$506$	0	0
$507$	−7.00000	−0.310881
$508$	0	0
$509$	33.4164	1.48116	0.740578	−	0.671970i	$-0.234552\pi$
$509$	33.4164	1.48116	0.740578	+	0.671970i	$0.234552\pi$
$510$	0	0
$511$	2.94427	0.130247
$512$	0	0
$513$	−6.47214	−0.285752
$514$	0	0
$515$	−25.8885	−1.14079
$516$	0	0
$517$	0	0
$518$	0	0
$519$	8.47214	0.371885
$520$	0	0
$521$	6.58359	0.288432	0.144216	−	0.989546i	$-0.453934\pi$
$521$	6.58359	0.288432	0.144216	+	0.989546i	$0.453934\pi$
$522$	0	0
$523$	−37.3050	−1.63123	−0.815616	−	0.578594i	$-0.803602\pi$
$523$	−37.3050	−1.63123	−0.815616	+	0.578594i	$0.803602\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	28.9443	1.26083
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−26.8328	−1.16226
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	−20.0000	−0.863064
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	29.8885	1.28028
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−12.9443	−0.551445
$552$	0	0
$553$	12.9443	0.550446
$554$	0	0
$555$	21.8885	0.929117
$556$	0	0
$557$	−34.9443	−1.48064	−0.740318	−	0.672257i	$-0.765325\pi$
$557$	−34.9443	−1.48064	−0.740318	+	0.672257i	$0.765325\pi$
$558$	0	0
$559$	−57.8885	−2.44842
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.5279	−0.738711	−0.369356	−	0.929288i	$-0.620422\pi$
$563$	−17.5279	−0.738711	−0.369356	+	0.929288i	$0.620422\pi$
$564$	0	0
$565$	−2.11146	−0.0888296
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	27.8885	1.16915	0.584574	−	0.811340i	$-0.301262\pi$
$569$	27.8885	1.16915	0.584574	+	0.811340i	$0.301262\pi$
$570$	0	0
$571$	−4.94427	−0.206911	−0.103456	−	0.994634i	$-0.532990\pi$
$571$	−4.94427	−0.206911	−0.103456	+	0.994634i	$0.532990\pi$
$572$	0	0
$573$	−20.9443	−0.874960
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	14.9443	0.622138	0.311069	−	0.950387i	$-0.399313\pi$
$577$	14.9443	0.622138	0.311069	+	0.950387i	$0.399313\pi$
$578$	0	0
$579$	23.8885	0.992774
$580$	0	0
$581$	−6.47214	−0.268509
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.94427	0.369800
$586$	0	0
$587$	−16.9443	−0.699365	−0.349682	−	0.936868i	$-0.613711\pi$
$587$	−16.9443	−0.699365	−0.349682	+	0.936868i	$0.613711\pi$
$588$	0	0
$589$	−41.8885	−1.72599
$590$	0	0
$591$	−2.94427	−0.121111
$592$	0	0
$593$	24.8328	1.01976	0.509881	−	0.860245i	$-0.329690\pi$
$593$	24.8328	1.01976	0.509881	+	0.860245i	$0.329690\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	0	0
$597$	20.9443	0.857192
$598$	0	0
$599$	1.88854	0.0771638	0.0385819	−	0.999255i	$-0.487716\pi$
$599$	1.88854	0.0771638	0.0385819	+	0.999255i	$0.487716\pi$
$600$	0	0
$601$	−12.8328	−0.523461	−0.261731	−	0.965141i	$-0.584293\pi$
$601$	−12.8328	−0.523461	−0.261731	+	0.965141i	$0.584293\pi$
$602$	0	0
$603$	12.9443	0.527132
$604$	0	0
$605$	−22.0000	−0.894427
$606$	0	0
$607$	−16.3607	−0.664060	−0.332030	−	0.943269i	$-0.607733\pi$
$607$	−16.3607	−0.664060	−0.332030	+	0.943269i	$0.607733\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	−28.9443	−1.17096
$612$	0	0
$613$	−12.8328	−0.518313	−0.259156	−	0.965835i	$-0.583444\pi$
$613$	−12.8328	−0.518313	−0.259156	+	0.965835i	$0.583444\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	25.5279	1.02605	0.513026	−	0.858373i	$-0.328525\pi$
$619$	25.5279	1.02605	0.513026	+	0.858373i	$0.328525\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	17.4164	0.697774
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	48.9443	1.95154
$630$	0	0
$631$	28.9443	1.15225	0.576127	−	0.817360i	$-0.304564\pi$
$631$	28.9443	1.15225	0.576127	+	0.817360i	$0.304564\pi$
$632$	0	0
$633$	16.9443	0.673474
$634$	0	0
$635$	−6.11146	−0.242526
$636$	0	0
$637$	4.47214	0.177192
$638$	0	0
$639$	12.9443	0.512067
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	1.52786	0.0602531	0.0301265	−	0.999546i	$-0.490409\pi$
$643$	1.52786	0.0602531	0.0301265	+	0.999546i	$0.490409\pi$
$644$	0	0
$645$	25.8885	1.01936
$646$	0	0
$647$	32.3607	1.27223	0.636115	−	0.771594i	$-0.280540\pi$
$647$	32.3607	1.27223	0.636115	+	0.771594i	$0.280540\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.47214	−0.253663
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	0	0
$657$	−2.94427	−0.114867
$658$	0	0
$659$	6.83282	0.266169	0.133084	−	0.991105i	$-0.457512\pi$
$659$	6.83282	0.266169	0.133084	+	0.991105i	$0.457512\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	0	0
$663$	20.0000	0.776736
$664$	0	0
$665$	−12.9443	−0.501957
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	9.52786	0.368369
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−39.8885	−1.53304	−0.766521	−	0.642220i	$-0.778014\pi$
$677$	−39.8885	−1.53304	−0.766521	+	0.642220i	$0.778014\pi$
$678$	0	0
$679$	−8.47214	−0.325131
$680$	0	0
$681$	14.4721	0.554573
$682$	0	0
$683$	0.944272	0.0361316	0.0180658	−	0.999837i	$-0.494249\pi$
$683$	0.944272	0.0361316	0.0180658	+	0.999837i	$0.494249\pi$
$684$	0	0
$685$	−28.0000	−1.06983
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	31.0557	1.18313
$690$	0	0
$691$	32.9443	1.25326	0.626630	−	0.779317i	$-0.284434\pi$
$691$	32.9443	1.25326	0.626630	+	0.779317i	$0.284434\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.8885	0.678551
$696$	0	0
$697$	26.8328	1.01637
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	32.8328	1.24008	0.620039	−	0.784571i	$-0.287117\pi$
$701$	32.8328	1.24008	0.620039	+	0.784571i	$0.287117\pi$
$702$	0	0
$703$	−70.8328	−2.67151
$704$	0	0
$705$	12.9443	0.487509
$706$	0	0
$707$	−17.4164	−0.655011
$708$	0	0
$709$	−10.9443	−0.411021	−0.205510	−	0.978655i	$-0.565885\pi$
$709$	−10.9443	−0.411021	−0.205510	+	0.978655i	$0.565885\pi$
$710$	0	0
$711$	−12.9443	−0.485448
$712$	0	0
$713$	−6.47214	−0.242383
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.05573	0.114118
$718$	0	0
$719$	14.4721	0.539720	0.269860	−	0.962900i	$-0.413023\pi$
$719$	14.4721	0.539720	0.269860	+	0.962900i	$0.413023\pi$
$720$	0	0
$721$	12.9443	0.482070
$722$	0	0
$723$	−11.5279	−0.428726
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−20.9443	−0.776780	−0.388390	−	0.921495i	$-0.626969\pi$
$727$	−20.9443	−0.776780	−0.388390	+	0.921495i	$0.626969\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	57.8885	2.14109
$732$	0	0
$733$	−33.0557	−1.22094	−0.610471	−	0.792039i	$-0.709020\pi$
$733$	−33.0557	−1.22094	−0.610471	+	0.792039i	$0.709020\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−8.94427	−0.329020	−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427	−0.329020	−0.164510	+	0.986375i	$0.552604\pi$
$740$	0	0
$741$	−28.9443	−1.06329
$742$	0	0
$743$	−30.8328	−1.13115	−0.565573	−	0.824698i	$-0.691345\pi$
$743$	−30.8328	−1.13115	−0.565573	+	0.824698i	$0.691345\pi$
$744$	0	0
$745$	−37.8885	−1.38813
$746$	0	0
$747$	6.47214	0.236803
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	17.8885	0.652762	0.326381	−	0.945238i	$-0.394171\pi$
$751$	17.8885	0.652762	0.326381	+	0.945238i	$0.394171\pi$
$752$	0	0
$753$	−14.4721	−0.527394
$754$	0	0
$755$	25.8885	0.942181
$756$	0	0
$757$	−25.0557	−0.910666	−0.455333	−	0.890321i	$-0.650480\pi$
$757$	−25.0557	−0.910666	−0.455333	+	0.890321i	$0.650480\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.9443	1.41173	0.705864	−	0.708347i	$-0.250559\pi$
$761$	38.9443	1.41173	0.705864	+	0.708347i	$0.250559\pi$
$762$	0	0
$763$	−14.9443	−0.541019
$764$	0	0
$765$	−8.94427	−0.323381
$766$	0	0
$767$	−17.8885	−0.645918
$768$	0	0
$769$	−14.3607	−0.517859	−0.258930	−	0.965896i	$-0.583370\pi$
$769$	−14.3607	−0.517859	−0.258930	+	0.965896i	$0.583370\pi$
$770$	0	0
$771$	−14.9443	−0.538205
$772$	0	0
$773$	22.9443	0.825248	0.412624	−	0.910901i	$-0.364612\pi$
$773$	22.9443	0.825248	0.412624	+	0.910901i	$0.364612\pi$
$774$	0	0
$775$	6.47214	0.232486
$776$	0	0
$777$	−10.9443	−0.392624
$778$	0	0
$779$	−38.8328	−1.39133
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	29.8885	1.06677
$786$	0	0
$787$	27.4164	0.977289	0.488645	−	0.872483i	$-0.337491\pi$
$787$	27.4164	0.977289	0.488645	+	0.872483i	$0.337491\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	1.05573	0.0375374
$792$	0	0
$793$	−26.8328	−0.952861
$794$	0	0
$795$	−13.8885	−0.492576
$796$	0	0
$797$	24.8328	0.879623	0.439812	−	0.898090i	$-0.355045\pi$
$797$	24.8328	0.879623	0.439812	+	0.898090i	$0.355045\pi$
$798$	0	0
$799$	28.9443	1.02397
$800$	0	0
$801$	−17.4164	−0.615379
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	−7.52786	−0.264993
$808$	0	0
$809$	0.111456	0.00391859	0.00195930	−	0.999998i	$-0.499376\pi$
$809$	0.111456	0.00391859	0.00195930	+	0.999998i	$0.499376\pi$
$810$	0	0
$811$	2.11146	0.0741433	0.0370716	−	0.999313i	$-0.488197\pi$
$811$	2.11146	0.0741433	0.0370716	+	0.999313i	$0.488197\pi$
$812$	0	0
$813$	−11.4164	−0.400391
$814$	0	0
$815$	−17.8885	−0.626608
$816$	0	0
$817$	−83.7771	−2.93099
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	36.8328	1.28547	0.642737	−	0.766087i	$-0.277799\pi$
$821$	36.8328	1.28547	0.642737	+	0.766087i	$0.277799\pi$
$822$	0	0
$823$	−3.05573	−0.106516	−0.0532580	−	0.998581i	$-0.516961\pi$
$823$	−3.05573	−0.106516	−0.0532580	+	0.998581i	$0.516961\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.88854	0.0656711	0.0328356	−	0.999461i	$-0.489546\pi$
$827$	1.88854	0.0656711	0.0328356	+	0.999461i	$0.489546\pi$
$828$	0	0
$829$	−13.4164	−0.465971	−0.232986	−	0.972480i	$-0.574849\pi$
$829$	−13.4164	−0.465971	−0.232986	+	0.972480i	$0.574849\pi$
$830$	0	0
$831$	−23.8885	−0.828684
$832$	0	0
$833$	−4.47214	−0.154950
$834$	0	0
$835$	−22.8328	−0.790162
$836$	0	0
$837$	6.47214	0.223710
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	14.0000	0.481615
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	9.52786	0.326995
$850$	0	0
$851$	−10.9443	−0.375165
$852$	0	0
$853$	38.3607	1.31344	0.656722	−	0.754132i	$-0.271942\pi$
$853$	38.3607	1.31344	0.656722	+	0.754132i	$0.271942\pi$
$854$	0	0
$855$	12.9443	0.442685
$856$	0	0
$857$	−12.1115	−0.413719	−0.206860	−	0.978371i	$-0.566324\pi$
$857$	−12.1115	−0.413719	−0.206860	+	0.978371i	$0.566324\pi$
$858$	0	0
$859$	−15.0557	−0.513695	−0.256847	−	0.966452i	$-0.582684\pi$
$859$	−15.0557	−0.513695	−0.256847	+	0.966452i	$0.582684\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−3.05573	−0.104018	−0.0520091	−	0.998647i	$-0.516562\pi$
$863$	−3.05573	−0.104018	−0.0520091	+	0.998647i	$0.516562\pi$
$864$	0	0
$865$	−16.9443	−0.576123
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	57.8885	1.96148
$872$	0	0
$873$	8.47214	0.286738
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	54.7214	1.84781	0.923905	−	0.382623i	$-0.124979\pi$
$877$	54.7214	1.84781	0.923905	+	0.382623i	$0.124979\pi$
$878$	0	0
$879$	7.88854	0.266074
$880$	0	0
$881$	−22.3607	−0.753350	−0.376675	−	0.926345i	$-0.622933\pi$
$881$	−22.3607	−0.753350	−0.376675	+	0.926345i	$0.622933\pi$
$882$	0	0
$883$	50.8328	1.71066	0.855330	−	0.518083i	$-0.173354\pi$
$883$	50.8328	1.71066	0.855330	+	0.518083i	$0.173354\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	14.4721	0.485927	0.242963	−	0.970035i	$-0.421881\pi$
$887$	14.4721	0.485927	0.242963	+	0.970035i	$0.421881\pi$
$888$	0	0
$889$	3.05573	0.102486
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−41.8885	−1.40175
$894$	0	0
$895$	40.0000	1.33705
$896$	0	0
$897$	−4.47214	−0.149320
$898$	0	0
$899$	12.9443	0.431716
$900$	0	0
$901$	−31.0557	−1.03462
$902$	0	0
$903$	−12.9443	−0.430758
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	6.11146	0.202928	0.101464	−	0.994839i	$-0.467647\pi$
$907$	6.11146	0.202928	0.101464	+	0.994839i	$0.467647\pi$
$908$	0	0
$909$	17.4164	0.577666
$910$	0	0
$911$	−40.7214	−1.34916	−0.674579	−	0.738202i	$-0.735675\pi$
$911$	−40.7214	−1.34916	−0.674579	+	0.738202i	$0.735675\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	12.0000	0.396708
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	−52.9443	−1.74647	−0.873235	−	0.487299i	$-0.837982\pi$
$919$	−52.9443	−1.74647	−0.873235	+	0.487299i	$0.837982\pi$
$920$	0	0
$921$	7.05573	0.232494
$922$	0	0
$923$	57.8885	1.90542
$924$	0	0
$925$	10.9443	0.359845
$926$	0	0
$927$	−12.9443	−0.425146
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	6.47214	0.212116
$932$	0	0
$933$	6.47214	0.211888
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.2492	−1.31488	−0.657442	−	0.753505i	$-0.728362\pi$
$937$	−40.2492	−1.31488	−0.657442	+	0.753505i	$0.728362\pi$
$938$	0	0
$939$	−13.4164	−0.437828
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	16.9443	0.550615	0.275307	−	0.961356i	$-0.411220\pi$
$947$	16.9443	0.550615	0.275307	+	0.961356i	$0.411220\pi$
$948$	0	0
$949$	−13.1672	−0.427425
$950$	0	0
$951$	30.9443	1.00344
$952$	0	0
$953$	−26.9443	−0.872811	−0.436405	−	0.899750i	$-0.643749\pi$
$953$	−26.9443	−0.872811	−0.436405	+	0.899750i	$0.643749\pi$
$954$	0	0
$955$	41.8885	1.35548
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	−47.7771	−1.53800
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	28.9443	0.929824
$970$	0	0
$971$	−3.41641	−0.109638	−0.0548189	−	0.998496i	$-0.517458\pi$
$971$	−3.41641	−0.109638	−0.0548189	+	0.998496i	$0.517458\pi$
$972$	0	0
$973$	−8.94427	−0.286740
$974$	0	0
$975$	4.47214	0.143223
$976$	0	0
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	14.9443	0.477134
$982$	0	0
$983$	57.8885	1.84636	0.923179	−	0.384371i	$-0.125581\pi$
$983$	57.8885	1.84636	0.923179	+	0.384371i	$0.125581\pi$
$984$	0	0
$985$	5.88854	0.187625
$986$	0	0
$987$	−6.47214	−0.206010
$988$	0	0
$989$	−12.9443	−0.411604
$990$	0	0
$991$	−43.0557	−1.36771	−0.683855	−	0.729618i	$-0.739698\pi$
$991$	−43.0557	−1.36771	−0.683855	+	0.729618i	$0.739698\pi$
$992$	0	0
$993$	21.8885	0.694612
$994$	0	0
$995$	−41.8885	−1.32796
$996$	0	0
$997$	−1.63932	−0.0519178	−0.0259589	−	0.999663i	$-0.508264\pi$
$997$	−1.63932	−0.0519178	−0.0259589	+	0.999663i	$0.508264\pi$
$998$	0	0
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bd.1.2		2
4.3	odd	2		966.2.a.p.1.2	✓	2
12.11	even	2		2898.2.a.v.1.2		2
28.27	even	2		6762.2.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.p.1.2	✓	2	4.3	odd	2
2898.2.a.v.1.2		2	12.11	even	2
6762.2.a.bz.1.1		2	28.27	even	2
7728.2.a.bd.1.2		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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} -2.00000 q^{15} -4.47214 q^{17} +6.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.47214 q^{31} -2.00000 q^{35} -10.9443 q^{37} -4.47214 q^{39} -6.00000 q^{41} -12.9443 q^{43} +2.00000 q^{45} -6.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} +6.94427 q^{53} -6.47214 q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} +8.94427 q^{65} +12.9443 q^{67} -1.00000 q^{69} +12.9443 q^{71} -2.94427 q^{73} +1.00000 q^{75} -12.9443 q^{79} +1.00000 q^{81} +6.47214 q^{83} -8.94427 q^{85} +2.00000 q^{87} -17.4164 q^{89} -4.47214 q^{91} +6.47214 q^{93} +12.9443 q^{95} +8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 + 4 * q^19 + 2 * q^21 + 2 * q^23 - 2 * q^25 - 2 * q^27 - 4 * q^29 - 4 * q^31 - 4 * q^35 - 4 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 - 4 * q^47 + 2 * q^49 - 4 * q^53 - 4 * q^57 - 8 * q^59 - 12 * q^61 - 2 * q^63 + 8 * q^67 - 2 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^87 - 8 * q^89 + 4 * q^93 + 8 * q^95 + 8 * q^97

Introduction

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