LMFDB - Embedded newform 882.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,2,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, 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("882.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	882.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:	$7.04280545828$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{8} -1.41421 q^{10} -4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -7.07107 q^{17} +5.65685 q^{19} +1.41421 q^{20} +4.00000 q^{22} -8.00000 q^{23} -3.00000 q^{25} +4.24264 q^{26} -2.00000 q^{29} -1.00000 q^{32} +7.07107 q^{34} +4.00000 q^{37} -5.65685 q^{38} -1.41421 q^{40} +9.89949 q^{41} -4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +5.65685 q^{47} +3.00000 q^{50} -4.24264 q^{52} -4.00000 q^{53} -5.65685 q^{55} +2.00000 q^{58} -11.3137 q^{59} -1.41421 q^{61} +1.00000 q^{64} -6.00000 q^{65} -12.0000 q^{67} -7.07107 q^{68} +15.5563 q^{73} -4.00000 q^{74} +5.65685 q^{76} -16.0000 q^{79} +1.41421 q^{80} -9.89949 q^{82} -5.65685 q^{83} -10.0000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +7.07107 q^{89} -8.00000 q^{92} -5.65685 q^{94} +8.00000 q^{95} -7.07107 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 8 q^{11} + 2 q^{16} + 8 q^{22} - 16 q^{23} - 6 q^{25} - 4 q^{29} - 2 q^{32} + 8 q^{37} - 8 q^{43} - 8 q^{44} + 16 q^{46} + 6 q^{50} - 8 q^{53} + 4 q^{58} + 2 q^{64} - 12 q^{65} - 24 q^{67} - 8 q^{74} - 32 q^{79} - 20 q^{85} + 8 q^{86} + 8 q^{88} - 16 q^{92} + 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 8 * q^11 + 2 * q^16 + 8 * q^22 - 16 * q^23 - 6 * q^25 - 4 * q^29 - 2 * q^32 + 8 * q^37 - 8 * q^43 - 8 * q^44 + 16 * q^46 + 6 * q^50 - 8 * q^53 + 4 * q^58 + 2 * q^64 - 12 * q^65 - 24 * q^67 - 8 * q^74 - 32 * q^79 - 20 * q^85 + 8 * q^86 + 8 * q^88 - 16 * q^92 + 16 * q^95

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.41421	−0.447214
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.07107	−1.71499	−0.857493	−	0.514496i	$-0.827979\pi$
$17$	−7.07107	−1.71499	−0.857493	+	0.514496i	$0.827979\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	1.41421	0.316228
$21$	0	0
$22$	4.00000	0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	4.24264	0.832050
$27$	0	0
$28$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.07107	1.21268
$35$	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	$0.393356\pi$
$38$	−5.65685	−0.917663
$39$	0	0
$40$	−1.41421	−0.223607
$41$	9.89949	1.54604	0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949	1.54604	0.773021	+	0.634381i	$0.218745\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	0	0
$50$	3.00000	0.424264
$51$	0	0
$52$	−4.24264	−0.588348
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	0	0
$61$	−1.41421	−0.181071	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421	−0.181071	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−7.07107	−0.857493
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	15.5563	1.82073	0.910366	−	0.413803i	$-0.135800\pi$
$73$	15.5563	1.82073	0.910366	+	0.413803i	$0.135800\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	5.65685	0.648886
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.41421	0.158114
$81$	0	0
$82$	−9.89949	−1.09322
$83$	−5.65685	−0.620920	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685	−0.620920	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	4.00000	0.431331
$87$	0	0
$88$	4.00000	0.426401
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	0	0
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−5.65685	−0.583460
$95$	8.00000	0.820783
$96$	0	0
$97$	−7.07107	−0.717958	−0.358979	−	0.933346i	$-0.616875\pi$
$97$	−7.07107	−0.717958	−0.358979	+	0.933346i	$0.616875\pi$
$98$	0	0
$99$	0	0
$100$	−3.00000	−0.300000
$101$	12.7279	1.26648	0.633238	−	0.773957i	$-0.281726\pi$
$101$	12.7279	1.26648	0.633238	+	0.773957i	$0.281726\pi$
$102$	0	0
$103$	−5.65685	−0.557386	−0.278693	−	0.960380i	$-0.589901\pi$
$103$	−5.65685	−0.557386	−0.278693	+	0.960380i	$0.589901\pi$
$104$	4.24264	0.416025
$105$	0	0
$106$	4.00000	0.388514
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	5.65685	0.539360
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	−2.00000	−0.185695
$117$	0	0
$118$	11.3137	1.04151
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	1.41421	0.128037
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	7.07107	0.606339
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	5.65685	0.479808	0.239904	−	0.970797i	$-0.422884\pi$
$139$	5.65685	0.479808	0.239904	+	0.970797i	$0.422884\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.9706	1.41915
$144$	0	0
$145$	−2.82843	−0.234888
$146$	−15.5563	−1.28745
$147$	0	0
$148$	4.00000	0.328798
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−5.65685	−0.458831
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.3848	1.46726	0.733632	−	0.679546i	$-0.237823\pi$
$157$	18.3848	1.46726	0.733632	+	0.679546i	$0.237823\pi$
$158$	16.0000	1.27289
$159$	0	0
$160$	−1.41421	−0.111803
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	9.89949	0.773021
$165$	0	0
$166$	5.65685	0.439057
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	10.0000	0.766965
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−12.7279	−0.967686	−0.483843	−	0.875155i	$-0.660759\pi$
$173$	−12.7279	−0.967686	−0.483843	+	0.875155i	$0.660759\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−7.07107	−0.529999
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−12.7279	−0.946059	−0.473029	−	0.881047i	$-0.656840\pi$
$181$	−12.7279	−0.946059	−0.473029	+	0.881047i	$0.656840\pi$
$182$	0	0
$183$	0	0
$184$	8.00000	0.589768
$185$	5.65685	0.415900
$186$	0	0
$187$	28.2843	2.06835
$188$	5.65685	0.412568
$189$	0	0
$190$	−8.00000	−0.580381
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	7.07107	0.507673
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−12.7279	−0.895533
$203$	0	0
$204$	0	0
$205$	14.0000	0.977802
$206$	5.65685	0.394132
$207$	0	0
$208$	−4.24264	−0.294174
$209$	−22.6274	−1.56517
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	−4.00000	−0.270914
$219$	0	0
$220$	−5.65685	−0.381385
$221$	30.0000	2.01802
$222$	0	0
$223$	16.9706	1.13643	0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706	1.13643	0.568216	+	0.822879i	$0.307634\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.9706	1.12638	0.563188	−	0.826329i	$-0.309575\pi$
$227$	16.9706	1.12638	0.563188	+	0.826329i	$0.309575\pi$
$228$	0	0
$229$	−12.7279	−0.841085	−0.420542	−	0.907273i	$-0.638160\pi$
$229$	−12.7279	−0.841085	−0.420542	+	0.907273i	$0.638160\pi$
$230$	11.3137	0.746004
$231$	0	0
$232$	2.00000	0.131306
$233$	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
$235$	8.00000	0.521862
$236$	−11.3137	−0.736460
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−4.24264	−0.273293	−0.136646	−	0.990620i	$-0.543632\pi$
$241$	−4.24264	−0.273293	−0.136646	+	0.990620i	$0.543632\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−1.41421	−0.0905357
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	0	0
$250$	11.3137	0.715542
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.7279	−0.793946	−0.396973	−	0.917830i	$-0.629939\pi$
$257$	−12.7279	−0.793946	−0.396973	+	0.917830i	$0.629939\pi$
$258$	0	0
$259$	0	0
$260$	−6.00000	−0.372104
$261$	0	0
$262$	16.9706	1.04844
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−1.41421	−0.0862261	−0.0431131	−	0.999070i	$-0.513728\pi$
$269$	−1.41421	−0.0862261	−0.0431131	+	0.999070i	$0.513728\pi$
$270$	0	0
$271$	5.65685	0.343629	0.171815	−	0.985129i	$-0.445037\pi$
$271$	5.65685	0.343629	0.171815	+	0.985129i	$0.445037\pi$
$272$	−7.07107	−0.428746
$273$	0	0
$274$	6.00000	0.362473
$275$	12.0000	0.723627
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−5.65685	−0.339276
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−22.6274	−1.34506	−0.672530	−	0.740070i	$-0.734792\pi$
$283$	−22.6274	−1.34506	−0.672530	+	0.740070i	$0.734792\pi$
$284$	0	0
$285$	0	0
$286$	−16.9706	−1.00349
$287$	0	0
$288$	0	0
$289$	33.0000	1.94118
$290$	2.82843	0.166091
$291$	0	0
$292$	15.5563	0.910366
$293$	24.0416	1.40453	0.702264	−	0.711917i	$-0.252173\pi$
$293$	24.0416	1.40453	0.702264	+	0.711917i	$0.252173\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−20.0000	−1.15857
$299$	33.9411	1.96287
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	5.65685	0.324443
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−5.65685	−0.322854	−0.161427	−	0.986885i	$-0.551610\pi$
$307$	−5.65685	−0.322854	−0.161427	+	0.986885i	$0.551610\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.65685	−0.320771	−0.160385	−	0.987054i	$-0.551274\pi$
$311$	−5.65685	−0.320771	−0.160385	+	0.987054i	$0.551274\pi$
$312$	0	0
$313$	−21.2132	−1.19904	−0.599521	−	0.800359i	$-0.704642\pi$
$313$	−21.2132	−1.19904	−0.599521	+	0.800359i	$0.704642\pi$
$314$	−18.3848	−1.03751
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	1.41421	0.0790569
$321$	0	0
$322$	0	0
$323$	−40.0000	−2.22566
$324$	0	0
$325$	12.7279	0.706018
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−9.89949	−0.546608
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−5.65685	−0.310460
$333$	0	0
$334$	−11.3137	−0.619059
$335$	−16.9706	−0.927201
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	−5.00000	−0.271964
$339$	0	0
$340$	−10.0000	−0.542326
$341$	0	0
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	12.7279	0.684257
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	29.6985	1.58972	0.794862	−	0.606791i	$-0.207543\pi$
$349$	29.6985	1.58972	0.794862	+	0.606791i	$0.207543\pi$
$350$	0	0
$351$	0	0
$352$	4.00000	0.213201
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	0	0
$355$	0	0
$356$	7.07107	0.374766
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	12.7279	0.668965
$363$	0	0
$364$	0	0
$365$	22.0000	1.15153
$366$	0	0
$367$	−5.65685	−0.295285	−0.147643	−	0.989041i	$-0.547169\pi$
$367$	−5.65685	−0.295285	−0.147643	+	0.989041i	$0.547169\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	−5.65685	−0.294086
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−28.2843	−1.46254
$375$	0	0
$376$	−5.65685	−0.291730
$377$	8.48528	0.437014
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−7.07107	−0.358979
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	56.5685	2.86079
$392$	0	0
$393$	0	0
$394$	−4.00000	−0.201517
$395$	−22.6274	−1.13851
$396$	0	0
$397$	−7.07107	−0.354887	−0.177443	−	0.984131i	$-0.556783\pi$
$397$	−7.07107	−0.354887	−0.177443	+	0.984131i	$0.556783\pi$
$398$	16.9706	0.850657
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	12.7279	0.633238
$405$	0	0
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	21.2132	1.04893	0.524463	−	0.851433i	$-0.324266\pi$
$409$	21.2132	1.04893	0.524463	+	0.851433i	$0.324266\pi$
$410$	−14.0000	−0.691411
$411$	0	0
$412$	−5.65685	−0.278693
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	4.24264	0.208013
$417$	0	0
$418$	22.6274	1.10674
$419$	22.6274	1.10542	0.552711	−	0.833373i	$-0.313593\pi$
$419$	22.6274	1.10542	0.552711	+	0.833373i	$0.313593\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	4.00000	0.194257
$425$	21.2132	1.02899
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	5.65685	0.272798
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	4.24264	0.203888	0.101944	−	0.994790i	$-0.467494\pi$
$433$	4.24264	0.203888	0.101944	+	0.994790i	$0.467494\pi$
$434$	0	0
$435$	0	0
$436$	4.00000	0.191565
$437$	−45.2548	−2.16483
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	5.65685	0.269680
$441$	0	0
$442$	−30.0000	−1.42695
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	−16.9706	−0.803579
$447$	0	0
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	−39.5980	−1.86460
$452$	0	0
$453$	0	0
$454$	−16.9706	−0.796468
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	12.7279	0.594737
$459$	0	0
$460$	−11.3137	−0.527504
$461$	−1.41421	−0.0658665	−0.0329332	−	0.999458i	$-0.510485\pi$
$461$	−1.41421	−0.0658665	−0.0329332	+	0.999458i	$0.510485\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	5.65685	0.261768	0.130884	−	0.991398i	$-0.458218\pi$
$467$	5.65685	0.261768	0.130884	+	0.991398i	$0.458218\pi$
$468$	0	0
$469$	0	0
$470$	−8.00000	−0.369012
$471$	0	0
$472$	11.3137	0.520756
$473$	16.0000	0.735681
$474$	0	0
$475$	−16.9706	−0.778663
$476$	0	0
$477$	0	0
$478$	−24.0000	−1.09773
$479$	−28.2843	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	−16.9706	−0.773791
$482$	4.24264	0.193247
$483$	0	0
$484$	5.00000	0.227273
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	1.41421	0.0640184
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	14.1421	0.636930
$494$	24.0000	1.07981
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−11.3137	−0.505964
$501$	0	0
$502$	5.65685	0.252478
$503$	28.2843	1.26113	0.630567	−	0.776135i	$-0.282823\pi$
$503$	28.2843	1.26113	0.630567	+	0.776135i	$0.282823\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	−32.0000	−1.42257
$507$	0	0
$508$	−8.00000	−0.354943
$509$	32.5269	1.44173	0.720865	−	0.693075i	$-0.243745\pi$
$509$	32.5269	1.44173	0.720865	+	0.693075i	$0.243745\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.7279	0.561405
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−22.6274	−0.995153
$518$	0	0
$519$	0	0
$520$	6.00000	0.263117
$521$	1.41421	0.0619578	0.0309789	−	0.999520i	$-0.490138\pi$
$521$	1.41421	0.0619578	0.0309789	+	0.999520i	$0.490138\pi$
$522$	0	0
$523$	33.9411	1.48414	0.742071	−	0.670321i	$-0.233844\pi$
$523$	33.9411	1.48414	0.742071	+	0.670321i	$0.233844\pi$
$524$	−16.9706	−0.741362
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	5.65685	0.245718
$531$	0	0
$532$	0	0
$533$	−42.0000	−1.81922
$534$	0	0
$535$	5.65685	0.244567
$536$	12.0000	0.518321
$537$	0	0
$538$	1.41421	0.0609711
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−5.65685	−0.242983
$543$	0	0
$544$	7.07107	0.303170
$545$	5.65685	0.242313
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	−12.0000	−0.511682
$551$	−11.3137	−0.481980
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	5.65685	0.239904
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	−11.3137	−0.476816	−0.238408	−	0.971165i	$-0.576626\pi$
$563$	−11.3137	−0.476816	−0.238408	+	0.971165i	$0.576626\pi$
$564$	0	0
$565$	0	0
$566$	22.6274	0.951101
$567$	0	0
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	16.9706	0.709575
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	−12.7279	−0.529870	−0.264935	−	0.964266i	$-0.585351\pi$
$577$	−12.7279	−0.529870	−0.264935	+	0.964266i	$0.585351\pi$
$578$	−33.0000	−1.37262
$579$	0	0
$580$	−2.82843	−0.117444
$581$	0	0
$582$	0	0
$583$	16.0000	0.662652
$584$	−15.5563	−0.643726
$585$	0	0
$586$	−24.0416	−0.993151
$587$	−16.9706	−0.700450	−0.350225	−	0.936666i	$-0.613895\pi$
$587$	−16.9706	−0.700450	−0.350225	+	0.936666i	$0.613895\pi$
$588$	0	0
$589$	0	0
$590$	16.0000	0.658710
$591$	0	0
$592$	4.00000	0.164399
$593$	−35.3553	−1.45187	−0.725935	−	0.687763i	$-0.758593\pi$
$593$	−35.3553	−1.45187	−0.725935	+	0.687763i	$0.758593\pi$
$594$	0	0
$595$	0	0
$596$	20.0000	0.819232
$597$	0	0
$598$	−33.9411	−1.38796
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−4.24264	−0.173061	−0.0865305	−	0.996249i	$-0.527578\pi$
$601$	−4.24264	−0.173061	−0.0865305	+	0.996249i	$0.527578\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	7.07107	0.287480
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	−5.65685	−0.229416
$609$	0	0
$610$	2.00000	0.0809776
$611$	−24.0000	−0.970936
$612$	0	0
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	5.65685	0.228292
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	−11.3137	−0.454736	−0.227368	−	0.973809i	$-0.573012\pi$
$619$	−11.3137	−0.454736	−0.227368	+	0.973809i	$0.573012\pi$
$620$	0	0
$621$	0	0
$622$	5.65685	0.226819
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	21.2132	0.847850
$627$	0	0
$628$	18.3848	0.733632
$629$	−28.2843	−1.12777
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	16.0000	0.636446
$633$	0	0
$634$	28.0000	1.11202
$635$	−11.3137	−0.448971
$636$	0	0
$637$	0	0
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−1.41421	−0.0559017
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−11.3137	−0.446169	−0.223085	−	0.974799i	$-0.571613\pi$
$643$	−11.3137	−0.446169	−0.223085	+	0.974799i	$0.571613\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	33.9411	1.33436	0.667182	−	0.744895i	$-0.267500\pi$
$647$	33.9411	1.33436	0.667182	+	0.744895i	$0.267500\pi$
$648$	0	0
$649$	45.2548	1.77641
$650$	−12.7279	−0.499230
$651$	0	0
$652$	4.00000	0.156652
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	9.89949	0.386510
$657$	0	0
$658$	0	0
$659$	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
$661$	−4.24264	−0.165020	−0.0825098	−	0.996590i	$-0.526294\pi$
$661$	−4.24264	−0.165020	−0.0825098	+	0.996590i	$0.526294\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	5.65685	0.219529
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	11.3137	0.437741
$669$	0	0
$670$	16.9706	0.655630
$671$	5.65685	0.218380
$672$	0	0
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	5.00000	0.192308
$677$	4.24264	0.163058	0.0815290	−	0.996671i	$-0.474020\pi$
$677$	4.24264	0.163058	0.0815290	+	0.996671i	$0.474020\pi$
$678$	0	0
$679$	0	0
$680$	10.0000	0.383482
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−8.48528	−0.324206
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	16.9706	0.646527
$690$	0	0
$691$	50.9117	1.93677	0.968386	−	0.249457i	$-0.0802520\pi$
$691$	50.9117	1.93677	0.968386	+	0.249457i	$0.0802520\pi$
$692$	−12.7279	−0.483843
$693$	0	0
$694$	−12.0000	−0.455514
$695$	8.00000	0.303457
$696$	0	0
$697$	−70.0000	−2.65144
$698$	−29.6985	−1.12410
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	22.6274	0.853409
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−1.41421	−0.0532246
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	0	0
$712$	−7.07107	−0.264999
$713$	0	0
$714$	0	0
$715$	24.0000	0.897549
$716$	12.0000	0.448461
$717$	0	0
$718$	16.0000	0.597115
$719$	39.5980	1.47676	0.738378	−	0.674387i	$-0.235592\pi$
$719$	39.5980	1.47676	0.738378	+	0.674387i	$0.235592\pi$
$720$	0	0
$721$	0	0
$722$	−13.0000	−0.483810
$723$	0	0
$724$	−12.7279	−0.473029
$725$	6.00000	0.222834
$726$	0	0
$727$	−28.2843	−1.04901	−0.524503	−	0.851409i	$-0.675749\pi$
$727$	−28.2843	−1.04901	−0.524503	+	0.851409i	$0.675749\pi$
$728$	0	0
$729$	0	0
$730$	−22.0000	−0.814257
$731$	28.2843	1.04613
$732$	0	0
$733$	−12.7279	−0.470117	−0.235058	−	0.971981i	$-0.575528\pi$
$733$	−12.7279	−0.470117	−0.235058	+	0.971981i	$0.575528\pi$
$734$	5.65685	0.208798
$735$	0	0
$736$	8.00000	0.294884
$737$	48.0000	1.76810
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	5.65685	0.207950
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	28.2843	1.03626
$746$	−10.0000	−0.366126
$747$	0	0
$748$	28.2843	1.03418
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	5.65685	0.206284
$753$	0	0
$754$	−8.48528	−0.309016
$755$	−22.6274	−0.823496
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	−8.00000	−0.290191
$761$	−9.89949	−0.358856	−0.179428	−	0.983771i	$-0.557425\pi$
$761$	−9.89949	−0.358856	−0.179428	+	0.983771i	$0.557425\pi$
$762$	0	0
$763$	0	0
$764$	16.0000	0.578860
$765$	0	0
$766$	5.65685	0.204390
$767$	48.0000	1.73318
$768$	0	0
$769$	4.24264	0.152994	0.0764968	−	0.997070i	$-0.475627\pi$
$769$	4.24264	0.152994	0.0764968	+	0.997070i	$0.475627\pi$
$770$	0	0
$771$	0	0
$772$	14.0000	0.503871
$773$	32.5269	1.16991	0.584956	−	0.811065i	$-0.301112\pi$
$773$	32.5269	1.16991	0.584956	+	0.811065i	$0.301112\pi$
$774$	0	0
$775$	0	0
$776$	7.07107	0.253837
$777$	0	0
$778$	26.0000	0.932145
$779$	56.0000	2.00641
$780$	0	0
$781$	0	0
$782$	−56.5685	−2.02289
$783$	0	0
$784$	0	0
$785$	26.0000	0.927980
$786$	0	0
$787$	−5.65685	−0.201645	−0.100823	−	0.994904i	$-0.532147\pi$
$787$	−5.65685	−0.201645	−0.100823	+	0.994904i	$0.532147\pi$
$788$	4.00000	0.142494
$789$	0	0
$790$	22.6274	0.805047
$791$	0	0
$792$	0	0
$793$	6.00000	0.213066
$794$	7.07107	0.250943
$795$	0	0
$796$	−16.9706	−0.601506
$797$	12.7279	0.450846	0.225423	−	0.974261i	$-0.427624\pi$
$797$	12.7279	0.450846	0.225423	+	0.974261i	$0.427624\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	3.00000	0.106066
$801$	0	0
$802$	18.0000	0.635602
$803$	−62.2254	−2.19589
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−12.7279	−0.447767
$809$	40.0000	1.40633	0.703163	−	0.711029i	$-0.251771\pi$
$809$	40.0000	1.40633	0.703163	+	0.711029i	$0.251771\pi$
$810$	0	0
$811$	−33.9411	−1.19183	−0.595917	−	0.803046i	$-0.703211\pi$
$811$	−33.9411	−1.19183	−0.595917	+	0.803046i	$0.703211\pi$
$812$	0	0
$813$	0	0
$814$	16.0000	0.560800
$815$	5.65685	0.198151
$816$	0	0
$817$	−22.6274	−0.791633
$818$	−21.2132	−0.741702
$819$	0	0
$820$	14.0000	0.488901
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	5.65685	0.197066
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−43.8406	−1.52265	−0.761324	−	0.648372i	$-0.775450\pi$
$829$	−43.8406	−1.52265	−0.761324	+	0.648372i	$0.775450\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	−4.24264	−0.147087
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	−22.6274	−0.782586
$837$	0	0
$838$	−22.6274	−0.781651
$839$	−45.2548	−1.56237	−0.781185	−	0.624299i	$-0.785385\pi$
$839$	−45.2548	−1.56237	−0.781185	+	0.624299i	$0.785385\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	0	0
$844$	12.0000	0.413057
$845$	7.07107	0.243252
$846$	0	0
$847$	0	0
$848$	−4.00000	−0.137361
$849$	0	0
$850$	−21.2132	−0.727607
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−21.2132	−0.726326	−0.363163	−	0.931726i	$-0.618303\pi$
$853$	−21.2132	−0.726326	−0.363163	+	0.931726i	$0.618303\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	24.0416	0.821246	0.410623	−	0.911805i	$-0.365311\pi$
$857$	24.0416	0.821246	0.410623	+	0.911805i	$0.365311\pi$
$858$	0	0
$859$	−5.65685	−0.193009	−0.0965047	−	0.995333i	$-0.530766\pi$
$859$	−5.65685	−0.193009	−0.0965047	+	0.995333i	$0.530766\pi$
$860$	−5.65685	−0.192897
$861$	0	0
$862$	24.0000	0.817443
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−4.24264	−0.144171
$867$	0	0
$868$	0	0
$869$	64.0000	2.17105
$870$	0	0
$871$	50.9117	1.72508
$872$	−4.00000	−0.135457
$873$	0	0
$874$	45.2548	1.53077
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	33.9411	1.14546
$879$	0	0
$880$	−5.65685	−0.190693
$881$	21.2132	0.714691	0.357345	−	0.933972i	$-0.383682\pi$
$881$	21.2132	0.714691	0.357345	+	0.933972i	$0.383682\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	20.0000	0.671913
$887$	−22.6274	−0.759754	−0.379877	−	0.925037i	$-0.624034\pi$
$887$	−22.6274	−0.759754	−0.379877	+	0.925037i	$0.624034\pi$
$888$	0	0
$889$	0	0
$890$	−10.0000	−0.335201
$891$	0	0
$892$	16.9706	0.568216
$893$	32.0000	1.07084
$894$	0	0
$895$	16.9706	0.567263
$896$	0	0
$897$	0	0
$898$	−24.0000	−0.800890
$899$	0	0
$900$	0	0
$901$	28.2843	0.942286
$902$	39.5980	1.31847
$903$	0	0
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	16.9706	0.563188
$909$	0	0
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	22.6274	0.748858
$914$	6.00000	0.198462
$915$	0	0
$916$	−12.7279	−0.420542
$917$	0	0
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	11.3137	0.373002
$921$	0	0
$922$	1.41421	0.0465746
$923$	0	0
$924$	0	0
$925$	−12.0000	−0.394558
$926$	32.0000	1.05159
$927$	0	0
$928$	2.00000	0.0656532
$929$	18.3848	0.603185	0.301592	−	0.953437i	$-0.402482\pi$
$929$	18.3848	0.603185	0.301592	+	0.953437i	$0.402482\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	−5.65685	−0.185098
$935$	40.0000	1.30814
$936$	0	0
$937$	−15.5563	−0.508204	−0.254102	−	0.967177i	$-0.581780\pi$
$937$	−15.5563	−0.508204	−0.254102	+	0.967177i	$0.581780\pi$
$938$	0	0
$939$	0	0
$940$	8.00000	0.260931
$941$	26.8701	0.875939	0.437969	−	0.898990i	$-0.355698\pi$
$941$	26.8701	0.875939	0.437969	+	0.898990i	$0.355698\pi$
$942$	0	0
$943$	−79.1960	−2.57898
$944$	−11.3137	−0.368230
$945$	0	0
$946$	−16.0000	−0.520205
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−66.0000	−2.14245
$950$	16.9706	0.550598
$951$	0	0
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	22.6274	0.732206
$956$	24.0000	0.776215
$957$	0	0
$958$	28.2843	0.913823
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	16.9706	0.547153
$963$	0	0
$964$	−4.24264	−0.136646
$965$	19.7990	0.637352
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	10.0000	0.321081
$971$	−11.3137	−0.363074	−0.181537	−	0.983384i	$-0.558107\pi$
$971$	−11.3137	−0.363074	−0.181537	+	0.983384i	$0.558107\pi$
$972$	0	0
$973$	0	0
$974$	24.0000	0.769010
$975$	0	0
$976$	−1.41421	−0.0452679
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−28.2843	−0.903969
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	11.3137	0.360851	0.180426	−	0.983589i	$-0.442252\pi$
$983$	11.3137	0.360851	0.180426	+	0.983589i	$0.442252\pi$
$984$	0	0
$985$	5.65685	0.180242
$986$	−14.1421	−0.450377
$987$	0	0
$988$	−24.0000	−0.763542
$989$	32.0000	1.01754
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	7.07107	0.223943	0.111971	−	0.993711i	$-0.464283\pi$
$997$	7.07107	0.223943	0.111971	+	0.993711i	$0.464283\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.a.m.1.2	yes	2
3.2	odd	2		882.2.a.o.1.1	yes	2
4.3	odd	2		7056.2.a.cs.1.2		2
7.2	even	3		882.2.g.m.361.1		4
7.3	odd	6		882.2.g.m.667.2		4
7.4	even	3		882.2.g.m.667.1		4
7.5	odd	6		882.2.g.m.361.2		4
7.6	odd	2	inner	882.2.a.m.1.1	✓	2
12.11	even	2		7056.2.a.ci.1.1		2
21.2	odd	6		882.2.g.k.361.2		4
21.5	even	6		882.2.g.k.361.1		4
21.11	odd	6		882.2.g.k.667.2		4
21.17	even	6		882.2.g.k.667.1		4
21.20	even	2		882.2.a.o.1.2	yes	2
28.27	even	2		7056.2.a.cs.1.1		2
84.83	odd	2		7056.2.a.ci.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.a.m.1.1	✓	2	7.6	odd	2	inner
882.2.a.m.1.2	yes	2	1.1	even	1	trivial
882.2.a.o.1.1	yes	2	3.2	odd	2
882.2.a.o.1.2	yes	2	21.20	even	2
882.2.g.k.361.1		4	21.5	even	6
882.2.g.k.361.2		4	21.2	odd	6
882.2.g.k.667.1		4	21.17	even	6
882.2.g.k.667.2		4	21.11	odd	6
882.2.g.m.361.1		4	7.2	even	3
882.2.g.m.361.2		4	7.5	odd	6
882.2.g.m.667.1		4	7.4	even	3
882.2.g.m.667.2		4	7.3	odd	6
7056.2.a.ci.1.1		2	12.11	even	2
7056.2.a.ci.1.2		2	84.83	odd	2
7056.2.a.cs.1.1		2	28.27	even	2
7056.2.a.cs.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{8} -1.41421 q^{10} -4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -7.07107 q^{17} +5.65685 q^{19} +1.41421 q^{20} +4.00000 q^{22} -8.00000 q^{23} -3.00000 q^{25} +4.24264 q^{26} -2.00000 q^{29} -1.00000 q^{32} +7.07107 q^{34} +4.00000 q^{37} -5.65685 q^{38} -1.41421 q^{40} +9.89949 q^{41} -4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +5.65685 q^{47} +3.00000 q^{50} -4.24264 q^{52} -4.00000 q^{53} -5.65685 q^{55} +2.00000 q^{58} -11.3137 q^{59} -1.41421 q^{61} +1.00000 q^{64} -6.00000 q^{65} -12.0000 q^{67} -7.07107 q^{68} +15.5563 q^{73} -4.00000 q^{74} +5.65685 q^{76} -16.0000 q^{79} +1.41421 q^{80} -9.89949 q^{82} -5.65685 q^{83} -10.0000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +7.07107 q^{89} -8.00000 q^{92} -5.65685 q^{94} +8.00000 q^{95} -7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 8 q^{11} + 2 q^{16} + 8 q^{22} - 16 q^{23} - 6 q^{25} - 4 q^{29} - 2 q^{32} + 8 q^{37} - 8 q^{43} - 8 q^{44} + 16 q^{46} + 6 q^{50} - 8 q^{53} + 4 q^{58} + 2 q^{64} - 12 q^{65} - 24 q^{67} - 8 q^{74} - 32 q^{79} - 20 q^{85} + 8 q^{86} + 8 q^{88} - 16 q^{92} + 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 8 * q^11 + 2 * q^16 + 8 * q^22 - 16 * q^23 - 6 * q^25 - 4 * q^29 - 2 * q^32 + 8 * q^37 - 8 * q^43 - 8 * q^44 + 16 * q^46 + 6 * q^50 - 8 * q^53 + 4 * q^58 + 2 * q^64 - 12 * q^65 - 24 * q^67 - 8 * q^74 - 32 * q^79 - 20 * q^85 + 8 * q^86 + 8 * q^88 - 16 * q^92 + 16 * q^95

Introduction

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