LMFDB - Embedded newform 882.2.a.n.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:	$0$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} -2.82843 q^{10} +2.00000 q^{11} +1.00000 q^{16} -1.41421 q^{17} +7.07107 q^{19} +2.82843 q^{20} -2.00000 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +1.41421 q^{34} +10.0000 q^{37} -7.07107 q^{38} -2.82843 q^{40} -9.89949 q^{41} +2.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} +2.82843 q^{47} -3.00000 q^{50} +2.00000 q^{53} +5.65685 q^{55} +2.00000 q^{58} -1.41421 q^{59} -2.82843 q^{61} +8.48528 q^{62} +1.00000 q^{64} +12.0000 q^{67} -1.41421 q^{68} +12.0000 q^{71} +1.41421 q^{73} -10.0000 q^{74} +7.07107 q^{76} -4.00000 q^{79} +2.82843 q^{80} +9.89949 q^{82} +9.89949 q^{83} -4.00000 q^{85} -2.00000 q^{86} -2.00000 q^{88} -7.07107 q^{89} +4.00000 q^{92} -2.82843 q^{94} +20.0000 q^{95} -9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} + 2 q^{16} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 2 q^{32} + 20 q^{37} + 4 q^{43} + 4 q^{44} - 8 q^{46} - 6 q^{50} + 4 q^{53} + 4 q^{58} + 2 q^{64} + 24 q^{67} + 24 q^{71} - 20 q^{74} - 8 q^{79} - 8 q^{85} - 4 q^{86} - 4 q^{88} + 8 q^{92} + 40 q^{95}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^11 + 2 * q^16 - 4 * q^22 + 8 * q^23 + 6 * q^25 - 4 * q^29 - 2 * q^32 + 20 * q^37 + 4 * q^43 + 4 * q^44 - 8 * q^46 - 6 * q^50 + 4 * q^53 + 4 * q^58 + 2 * q^64 + 24 * q^67 + 24 * q^71 - 20 * q^74 - 8 * q^79 - 8 * q^85 - 4 * q^86 - 4 * q^88 + 8 * q^92 + 40 * 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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.82843	−0.894427
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	7.07107	1.62221	0.811107	−	0.584898i	$-0.198865\pi$
$19$	7.07107	1.62221	0.811107	+	0.584898i	$0.198865\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	−2.00000	−0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	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
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.41421	0.242536
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−7.07107	−1.14708
$39$	0	0
$40$	−2.82843	−0.447214
$41$	−9.89949	−1.54604	−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949	−1.54604	−0.773021	+	0.634381i	$0.781255\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	−3.00000	−0.424264
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	−2.82843	−0.362143	−0.181071	−	0.983470i	$-0.557957\pi$
$61$	−2.82843	−0.362143	−0.181071	+	0.983470i	$0.557957\pi$
$62$	8.48528	1.07763
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−1.41421	−0.171499
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	7.07107	0.811107
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	2.82843	0.316228
$81$	0	0
$82$	9.89949	1.09322
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−2.00000	−0.215666
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−2.82843	−0.291730
$95$	20.0000	2.05196
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	0	0
$100$	3.00000	0.300000
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−5.65685	−0.539360
$111$	0	0
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	11.3137	1.05501
$116$	−2.00000	−0.185695
$117$	0	0
$118$	1.41421	0.130189
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.82843	0.256074
$123$	0	0
$124$	−8.48528	−0.762001
$125$	−5.65685	−0.505964
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	12.7279	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	1.41421	0.121268
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−9.89949	−0.839664	−0.419832	−	0.907602i	$-0.637911\pi$
$139$	−9.89949	−0.839664	−0.419832	+	0.907602i	$0.637911\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	−5.65685	−0.469776
$146$	−1.41421	−0.117041
$147$	0	0
$148$	10.0000	0.821995
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−7.07107	−0.573539
$153$	0	0
$154$	0	0
$155$	−24.0000	−1.92773
$156$	0	0
$157$	11.3137	0.902932	0.451466	−	0.892288i	$-0.350901\pi$
$157$	11.3137	0.902932	0.451466	+	0.892288i	$0.350901\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−2.82843	−0.223607
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−9.89949	−0.773021
$165$	0	0
$166$	−9.89949	−0.768350
$167$	−19.7990	−1.53209	−0.766046	−	0.642786i	$-0.777779\pi$
$167$	−19.7990	−1.53209	−0.766046	+	0.642786i	$0.777779\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	4.00000	0.306786
$171$	0	0
$172$	2.00000	0.152499
$173$	−16.9706	−1.29025	−0.645124	−	0.764078i	$-0.723194\pi$
$173$	−16.9706	−1.29025	−0.645124	+	0.764078i	$0.723194\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	7.07107	0.529999
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	28.2843	2.07950
$186$	0	0
$187$	−2.82843	−0.206835
$188$	2.82843	0.206284
$189$	0	0
$190$	−20.0000	−1.45095
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	9.89949	0.710742
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−8.48528	−0.597022
$203$	0	0
$204$	0	0
$205$	−28.0000	−1.95560
$206$	2.82843	0.197066
$207$	0	0
$208$	0	0
$209$	14.1421	0.978232
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−4.00000	−0.273434
$215$	5.65685	0.385794
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	5.65685	0.381385
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−12.0000	−0.798228
$227$	−21.2132	−1.40797	−0.703985	−	0.710215i	$-0.748598\pi$
$227$	−21.2132	−1.40797	−0.703985	+	0.710215i	$0.748598\pi$
$228$	0	0
$229$	16.9706	1.12145	0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706	1.12145	0.560723	+	0.828003i	$0.310523\pi$
$230$	−11.3137	−0.746004
$231$	0	0
$232$	2.00000	0.131306
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	−1.41421	−0.0920575
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	21.2132	1.36646	0.683231	−	0.730202i	$-0.260574\pi$
$241$	21.2132	1.36646	0.683231	+	0.730202i	$0.260574\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−2.82843	−0.181071
$245$	0	0
$246$	0	0
$247$	0	0
$248$	8.48528	0.538816
$249$	0	0
$250$	5.65685	0.357771
$251$	9.89949	0.624851	0.312425	−	0.949942i	$-0.398859\pi$
$251$	9.89949	0.624851	0.312425	+	0.949942i	$0.398859\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.7279	0.793946	0.396973	−	0.917830i	$-0.370061\pi$
$257$	12.7279	0.793946	0.396973	+	0.917830i	$0.370061\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−12.7279	−0.786334
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	5.65685	0.347498
$266$	0	0
$267$	0	0
$268$	12.0000	0.733017
$269$	−11.3137	−0.689809	−0.344904	−	0.938638i	$-0.612089\pi$
$269$	−11.3137	−0.689809	−0.344904	+	0.938638i	$0.612089\pi$
$270$	0	0
$271$	−22.6274	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$271$	−22.6274	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$272$	−1.41421	−0.0857493
$273$	0	0
$274$	12.0000	0.724947
$275$	6.00000	0.361814
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	9.89949	0.593732
$279$	0	0
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	1.41421	0.0840663	0.0420331	−	0.999116i	$-0.486616\pi$
$283$	1.41421	0.0840663	0.0420331	+	0.999116i	$0.486616\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	5.65685	0.332182
$291$	0	0
$292$	1.41421	0.0827606
$293$	−19.7990	−1.15667	−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990	−1.15667	−0.578335	+	0.815800i	$0.696297\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−10.0000	−0.581238
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	7.07107	0.405554
$305$	−8.00000	−0.458079
$306$	0	0
$307$	9.89949	0.564994	0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949	0.564994	0.282497	+	0.959268i	$0.408837\pi$
$308$	0	0
$309$	0	0
$310$	24.0000	1.36311
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	−11.3137	−0.638470
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	2.82843	0.158114
$321$	0	0
$322$	0	0
$323$	−10.0000	−0.556415
$324$	0	0
$325$	0	0
$326$	−10.0000	−0.553849
$327$	0	0
$328$	9.89949	0.546608
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	9.89949	0.543305
$333$	0	0
$334$	19.7990	1.08335
$335$	33.9411	1.85440
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	−4.00000	−0.216930
$341$	−16.9706	−0.919007
$342$	0	0
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	16.9706	0.912343
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−1.41421	−0.0752710	−0.0376355	−	0.999292i	$-0.511983\pi$
$353$	−1.41421	−0.0752710	−0.0376355	+	0.999292i	$0.511983\pi$
$354$	0	0
$355$	33.9411	1.80141
$356$	−7.07107	−0.374766
$357$	0	0
$358$	12.0000	0.634220
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−28.2843	−1.47043
$371$	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	$0.416643\pi$
$374$	2.82843	0.146254
$375$	0	0
$376$	−2.82843	−0.145865
$377$	0	0
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	20.0000	1.02598
$381$	0	0
$382$	−4.00000	−0.204658
$383$	−36.7696	−1.87884	−0.939418	−	0.342773i	$-0.888634\pi$
$383$	−36.7696	−1.87884	−0.939418	+	0.342773i	$0.888634\pi$
$384$	0	0
$385$	0	0
$386$	16.0000	0.814379
$387$	0	0
$388$	−9.89949	−0.502571
$389$	−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
$391$	−5.65685	−0.286079
$392$	0	0
$393$	0	0
$394$	2.00000	0.100759
$395$	−11.3137	−0.569254
$396$	0	0
$397$	−22.6274	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$397$	−22.6274	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$398$	8.48528	0.425329
$399$	0	0
$400$	3.00000	0.150000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	8.48528	0.422159
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−38.1838	−1.88807	−0.944033	−	0.329851i	$-0.893001\pi$
$409$	−38.1838	−1.88807	−0.944033	+	0.329851i	$0.893001\pi$
$410$	28.0000	1.38282
$411$	0	0
$412$	−2.82843	−0.139347
$413$	0	0
$414$	0	0
$415$	28.0000	1.37447
$416$	0	0
$417$	0	0
$418$	−14.1421	−0.691714
$419$	−9.89949	−0.483622	−0.241811	−	0.970323i	$-0.577741\pi$
$419$	−9.89949	−0.483622	−0.241811	+	0.970323i	$0.577741\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−4.24264	−0.205798
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	−5.65685	−0.272798
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	29.6985	1.42722	0.713609	−	0.700544i	$-0.247059\pi$
$433$	29.6985	1.42722	0.713609	+	0.700544i	$0.247059\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	28.2843	1.35302
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	−5.65685	−0.269680
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−19.7990	−0.932298
$452$	12.0000	0.564433
$453$	0	0
$454$	21.2132	0.995585
$455$	0	0
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	−16.9706	−0.792982
$459$	0	0
$460$	11.3137	0.527504
$461$	39.5980	1.84426	0.922131	−	0.386878i	$-0.126447\pi$
$461$	39.5980	1.84426	0.922131	+	0.386878i	$0.126447\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	24.0000	1.11178
$467$	32.5269	1.50517	0.752583	−	0.658497i	$-0.228808\pi$
$467$	32.5269	1.50517	0.752583	+	0.658497i	$0.228808\pi$
$468$	0	0
$469$	0	0
$470$	−8.00000	−0.369012
$471$	0	0
$472$	1.41421	0.0650945
$473$	4.00000	0.183920
$474$	0	0
$475$	21.2132	0.973329
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−31.1127	−1.42158	−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127	−1.42158	−0.710788	+	0.703407i	$0.751661\pi$
$480$	0	0
$481$	0	0
$482$	−21.2132	−0.966235
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−28.0000	−1.27141
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	2.82843	0.128037
$489$	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
$493$	2.82843	0.127386
$494$	0	0
$495$	0	0
$496$	−8.48528	−0.381000
$497$	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	$0.528537\pi$
$500$	−5.65685	−0.252982
$501$	0	0
$502$	−9.89949	−0.441836
$503$	39.5980	1.76559	0.882793	−	0.469762i	$-0.155660\pi$
$503$	39.5980	1.76559	0.882793	+	0.469762i	$0.155660\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	−8.00000	−0.355643
$507$	0	0
$508$	16.0000	0.709885
$509$	22.6274	1.00294	0.501471	−	0.865174i	$-0.332792\pi$
$509$	22.6274	1.00294	0.501471	+	0.865174i	$0.332792\pi$
$510$	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
$517$	5.65685	0.248788
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	0	0
$523$	−12.7279	−0.556553	−0.278277	−	0.960501i	$-0.589763\pi$
$523$	−12.7279	−0.556553	−0.278277	+	0.960501i	$0.589763\pi$
$524$	12.7279	0.556022
$525$	0	0
$526$	12.0000	0.523225
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−5.65685	−0.245718
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	11.3137	0.489134
$536$	−12.0000	−0.518321
$537$	0	0
$538$	11.3137	0.487769
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	22.6274	0.971931
$543$	0	0
$544$	1.41421	0.0606339
$545$	−5.65685	−0.242313
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−6.00000	−0.255841
$551$	−14.1421	−0.602475
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−9.89949	−0.419832
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	16.0000	0.674919
$563$	−1.41421	−0.0596020	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$	−1.41421	−0.0596020	−0.0298010	+	0.999556i	$0.509487\pi$
$564$	0	0
$565$	33.9411	1.42791
$566$	−1.41421	−0.0594438
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	21.2132	0.883117	0.441559	−	0.897232i	$-0.354426\pi$
$577$	21.2132	0.883117	0.441559	+	0.897232i	$0.354426\pi$
$578$	15.0000	0.623918
$579$	0	0
$580$	−5.65685	−0.234888
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	−1.41421	−0.0585206
$585$	0	0
$586$	19.7990	0.817889
$587$	−29.6985	−1.22579	−0.612894	−	0.790165i	$-0.709995\pi$
$587$	−29.6985	−1.22579	−0.612894	+	0.790165i	$0.709995\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	4.00000	0.164677
$591$	0	0
$592$	10.0000	0.410997
$593$	−7.07107	−0.290374	−0.145187	−	0.989404i	$-0.546378\pi$
$593$	−7.07107	−0.290374	−0.145187	+	0.989404i	$0.546378\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	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
$601$	−29.6985	−1.21143	−0.605713	−	0.795683i	$-0.707112\pi$
$601$	−29.6985	−1.21143	−0.605713	+	0.795683i	$0.707112\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	−19.7990	−0.804943
$606$	0	0
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	−7.07107	−0.286770
$609$	0	0
$610$	8.00000	0.323911
$611$	0	0
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	−9.89949	−0.399511
$615$	0	0
$616$	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
$619$	−18.3848	−0.738947	−0.369473	−	0.929241i	$-0.620462\pi$
$619$	−18.3848	−0.738947	−0.369473	+	0.929241i	$0.620462\pi$
$620$	−24.0000	−0.963863
$621$	0	0
$622$	11.3137	0.453638
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	12.7279	0.508710
$627$	0	0
$628$	11.3137	0.451466
$629$	−14.1421	−0.563884
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	10.0000	0.397151
$635$	45.2548	1.79588
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	0	0
$640$	−2.82843	−0.111803
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−9.89949	−0.390398	−0.195199	−	0.980764i	$-0.562535\pi$
$643$	−9.89949	−0.390398	−0.195199	+	0.980764i	$0.562535\pi$
$644$	0	0
$645$	0	0
$646$	10.0000	0.393445
$647$	8.48528	0.333591	0.166795	−	0.985992i	$-0.446658\pi$
$647$	8.48528	0.333591	0.166795	+	0.985992i	$0.446658\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	0	0
$651$	0	0
$652$	10.0000	0.391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	36.0000	1.40664
$656$	−9.89949	−0.386510
$657$	0	0
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−8.48528	−0.330039	−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528	−0.330039	−0.165020	+	0.986290i	$0.552769\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	−9.89949	−0.384175
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	−19.7990	−0.766046
$669$	0	0
$670$	−33.9411	−1.31126
$671$	−5.65685	−0.218380
$672$	0	0
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−16.9706	−0.652232	−0.326116	−	0.945330i	$-0.605740\pi$
$677$	−16.9706	−0.652232	−0.326116	+	0.945330i	$0.605740\pi$
$678$	0	0
$679$	0	0
$680$	4.00000	0.153393
$681$	0	0
$682$	16.9706	0.649836
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−33.9411	−1.29682
$686$	0	0
$687$	0	0
$688$	2.00000	0.0762493
$689$	0	0
$690$	0	0
$691$	−12.7279	−0.484193	−0.242096	−	0.970252i	$-0.577835\pi$
$691$	−12.7279	−0.484193	−0.242096	+	0.970252i	$0.577835\pi$
$692$	−16.9706	−0.645124
$693$	0	0
$694$	−30.0000	−1.13878
$695$	−28.0000	−1.06210
$696$	0	0
$697$	14.0000	0.530288
$698$	0	0
$699$	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
$703$	70.7107	2.66690
$704$	2.00000	0.0753778
$705$	0	0
$706$	1.41421	0.0532246
$707$	0	0
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−33.9411	−1.27379
$711$	0	0
$712$	7.07107	0.264999
$713$	−33.9411	−1.27111
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	−32.0000	−1.19423
$719$	2.82843	0.105483	0.0527413	−	0.998608i	$-0.483204\pi$
$719$	2.82843	0.105483	0.0527413	+	0.998608i	$0.483204\pi$
$720$	0	0
$721$	0	0
$722$	−31.0000	−1.15370
$723$	0	0
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	19.7990	0.734304	0.367152	−	0.930161i	$-0.380333\pi$
$727$	19.7990	0.734304	0.367152	+	0.930161i	$0.380333\pi$
$728$	0	0
$729$	0	0
$730$	−4.00000	−0.148047
$731$	−2.82843	−0.104613
$732$	0	0
$733$	−42.4264	−1.56706	−0.783528	−	0.621357i	$-0.786582\pi$
$733$	−42.4264	−1.56706	−0.783528	+	0.621357i	$0.786582\pi$
$734$	28.2843	1.04399
$735$	0	0
$736$	−4.00000	−0.147442
$737$	24.0000	0.884051
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	28.2843	1.03975
$741$	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
$748$	−2.82843	−0.103418
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	2.82843	0.103142
$753$	0	0
$754$	0	0
$755$	−45.2548	−1.64699
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	26.0000	0.944363
$759$	0	0
$760$	−20.0000	−0.725476
$761$	−7.07107	−0.256326	−0.128163	−	0.991753i	$-0.540908\pi$
$761$	−7.07107	−0.256326	−0.128163	+	0.991753i	$0.540908\pi$
$762$	0	0
$763$	0	0
$764$	4.00000	0.144715
$765$	0	0
$766$	36.7696	1.32854
$767$	0	0
$768$	0	0
$769$	−29.6985	−1.07095	−0.535477	−	0.844550i	$-0.679868\pi$
$769$	−29.6985	−1.07095	−0.535477	+	0.844550i	$0.679868\pi$
$770$	0	0
$771$	0	0
$772$	−16.0000	−0.575853
$773$	48.0833	1.72943	0.864717	−	0.502259i	$-0.167498\pi$
$773$	48.0833	1.72943	0.864717	+	0.502259i	$0.167498\pi$
$774$	0	0
$775$	−25.4558	−0.914401
$776$	9.89949	0.355371
$777$	0	0
$778$	26.0000	0.932145
$779$	−70.0000	−2.50801
$780$	0	0
$781$	24.0000	0.858788
$782$	5.65685	0.202289
$783$	0	0
$784$	0	0
$785$	32.0000	1.14213
$786$	0	0
$787$	1.41421	0.0504113	0.0252056	−	0.999682i	$-0.491976\pi$
$787$	1.41421	0.0504113	0.0252056	+	0.999682i	$0.491976\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	11.3137	0.402524
$791$	0	0
$792$	0	0
$793$	0	0
$794$	22.6274	0.803017
$795$	0	0
$796$	−8.48528	−0.300753
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	−3.00000	−0.106066
$801$	0	0
$802$	−18.0000	−0.635602
$803$	2.82843	0.0998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−8.48528	−0.298511
$809$	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
$811$	29.6985	1.04285	0.521427	−	0.853296i	$-0.325400\pi$
$811$	29.6985	1.04285	0.521427	+	0.853296i	$0.325400\pi$
$812$	0	0
$813$	0	0
$814$	−20.0000	−0.701000
$815$	28.2843	0.990755
$816$	0	0
$817$	14.1421	0.494771
$818$	38.1838	1.33506
$819$	0	0
$820$	−28.0000	−0.977802
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	2.82843	0.0985329
$825$	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
$829$	31.1127	1.08059	0.540294	−	0.841476i	$-0.318313\pi$
$829$	31.1127	1.08059	0.540294	+	0.841476i	$0.318313\pi$
$830$	−28.0000	−0.971894
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−56.0000	−1.93796
$836$	14.1421	0.489116
$837$	0	0
$838$	9.89949	0.341972
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−30.0000	−1.03387
$843$	0	0
$844$	−12.0000	−0.413057
$845$	−36.7696	−1.26491
$846$	0	0
$847$	0	0
$848$	2.00000	0.0686803
$849$	0	0
$850$	4.24264	0.145521
$851$	40.0000	1.37118
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	18.3848	0.628012	0.314006	−	0.949421i	$-0.398329\pi$
$857$	18.3848	0.628012	0.314006	+	0.949421i	$0.398329\pi$
$858$	0	0
$859$	26.8701	0.916795	0.458397	−	0.888747i	$-0.348424\pi$
$859$	26.8701	0.916795	0.458397	+	0.888747i	$0.348424\pi$
$860$	5.65685	0.192897
$861$	0	0
$862$	12.0000	0.408722
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	−29.6985	−1.00920
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	0	0
$874$	−28.2843	−0.956730
$875$	0	0
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−16.9706	−0.572729
$879$	0	0
$880$	5.65685	0.190693
$881$	29.6985	1.00057	0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985	1.00057	0.500284	+	0.865862i	$0.333229\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	0	0
$889$	0	0
$890$	20.0000	0.670402
$891$	0	0
$892$	0	0
$893$	20.0000	0.669274
$894$	0	0
$895$	−33.9411	−1.13453
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	16.9706	0.566000
$900$	0	0
$901$	−2.82843	−0.0942286
$902$	19.7990	0.659234
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−21.2132	−0.703985
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	19.7990	0.655251
$914$	−24.0000	−0.793849
$915$	0	0
$916$	16.9706	0.560723
$917$	0	0
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−11.3137	−0.373002
$921$	0	0
$922$	−39.5980	−1.30409
$923$	0	0
$924$	0	0
$925$	30.0000	0.986394
$926$	−16.0000	−0.525793
$927$	0	0
$928$	2.00000	0.0656532
$929$	32.5269	1.06717	0.533587	−	0.845745i	$-0.320844\pi$
$929$	32.5269	1.06717	0.533587	+	0.845745i	$0.320844\pi$
$930$	0	0
$931$	0	0
$932$	−24.0000	−0.786146
$933$	0	0
$934$	−32.5269	−1.06431
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−9.89949	−0.323402	−0.161701	−	0.986840i	$-0.551698\pi$
$937$	−9.89949	−0.323402	−0.161701	+	0.986840i	$0.551698\pi$
$938$	0	0
$939$	0	0
$940$	8.00000	0.260931
$941$	−31.1127	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$941$	−31.1127	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$942$	0	0
$943$	−39.5980	−1.28949
$944$	−1.41421	−0.0460287
$945$	0	0
$946$	−4.00000	−0.130051
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	0	0
$950$	−21.2132	−0.688247
$951$	0	0
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	12.0000	0.388108
$957$	0	0
$958$	31.1127	1.00521
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	21.2132	0.683231
$965$	−45.2548	−1.45680
$966$	0	0
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	28.0000	0.899026
$971$	32.5269	1.04384	0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269	1.04384	0.521919	+	0.852995i	$0.325216\pi$
$972$	0	0
$973$	0	0
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−2.82843	−0.0905357
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	−14.1421	−0.451985
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	48.0833	1.53362	0.766809	−	0.641875i	$-0.221843\pi$
$983$	48.0833	1.53362	0.766809	+	0.641875i	$0.221843\pi$
$984$	0	0
$985$	−5.65685	−0.180242
$986$	−2.82843	−0.0900755
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.48528	0.269408
$993$	0	0
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	31.1127	0.985349	0.492675	−	0.870214i	$-0.336019\pi$
$997$	31.1127	0.985349	0.492675	+	0.870214i	$0.336019\pi$
$998$	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.n.1.2		2
3.2	odd	2		98.2.a.b.1.2	yes	2
4.3	odd	2		7056.2.a.cl.1.2		2
7.2	even	3		882.2.g.l.361.1		4
7.3	odd	6		882.2.g.l.667.2		4
7.4	even	3		882.2.g.l.667.1		4
7.5	odd	6		882.2.g.l.361.2		4
7.6	odd	2	inner	882.2.a.n.1.1		2
12.11	even	2		784.2.a.l.1.1		2
15.2	even	4		2450.2.c.v.99.3		4
15.8	even	4		2450.2.c.v.99.2		4
15.14	odd	2		2450.2.a.bj.1.1		2
21.2	odd	6		98.2.c.c.67.1		4
21.5	even	6		98.2.c.c.67.2		4
21.11	odd	6		98.2.c.c.79.1		4
21.17	even	6		98.2.c.c.79.2		4
21.20	even	2		98.2.a.b.1.1	✓	2
24.5	odd	2		3136.2.a.bn.1.1		2
24.11	even	2		3136.2.a.bm.1.2		2
28.27	even	2		7056.2.a.cl.1.1		2
84.11	even	6		784.2.i.m.177.2		4
84.23	even	6		784.2.i.m.753.2		4
84.47	odd	6		784.2.i.m.753.1		4
84.59	odd	6		784.2.i.m.177.1		4
84.83	odd	2		784.2.a.l.1.2		2
105.62	odd	4		2450.2.c.v.99.4		4
105.83	odd	4		2450.2.c.v.99.1		4
105.104	even	2		2450.2.a.bj.1.2		2
168.83	odd	2		3136.2.a.bm.1.1		2
168.125	even	2		3136.2.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.2.a.b.1.1	✓	2	21.20	even	2
98.2.a.b.1.2	yes	2	3.2	odd	2
98.2.c.c.67.1		4	21.2	odd	6
98.2.c.c.67.2		4	21.5	even	6
98.2.c.c.79.1		4	21.11	odd	6
98.2.c.c.79.2		4	21.17	even	6
784.2.a.l.1.1		2	12.11	even	2
784.2.a.l.1.2		2	84.83	odd	2
784.2.i.m.177.1		4	84.59	odd	6
784.2.i.m.177.2		4	84.11	even	6
784.2.i.m.753.1		4	84.47	odd	6
784.2.i.m.753.2		4	84.23	even	6
882.2.a.n.1.1		2	7.6	odd	2	inner
882.2.a.n.1.2		2	1.1	even	1	trivial
882.2.g.l.361.1		4	7.2	even	3
882.2.g.l.361.2		4	7.5	odd	6
882.2.g.l.667.1		4	7.4	even	3
882.2.g.l.667.2		4	7.3	odd	6
2450.2.a.bj.1.1		2	15.14	odd	2
2450.2.a.bj.1.2		2	105.104	even	2
2450.2.c.v.99.1		4	105.83	odd	4
2450.2.c.v.99.2		4	15.8	even	4
2450.2.c.v.99.3		4	15.2	even	4
2450.2.c.v.99.4		4	105.62	odd	4
3136.2.a.bm.1.1		2	168.83	odd	2
3136.2.a.bm.1.2		2	24.11	even	2
3136.2.a.bn.1.1		2	24.5	odd	2
3136.2.a.bn.1.2		2	168.125	even	2
7056.2.a.cl.1.1		2	28.27	even	2
7056.2.a.cl.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} -2.82843 q^{10} +2.00000 q^{11} +1.00000 q^{16} -1.41421 q^{17} +7.07107 q^{19} +2.82843 q^{20} -2.00000 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +1.41421 q^{34} +10.0000 q^{37} -7.07107 q^{38} -2.82843 q^{40} -9.89949 q^{41} +2.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} +2.82843 q^{47} -3.00000 q^{50} +2.00000 q^{53} +5.65685 q^{55} +2.00000 q^{58} -1.41421 q^{59} -2.82843 q^{61} +8.48528 q^{62} +1.00000 q^{64} +12.0000 q^{67} -1.41421 q^{68} +12.0000 q^{71} +1.41421 q^{73} -10.0000 q^{74} +7.07107 q^{76} -4.00000 q^{79} +2.82843 q^{80} +9.89949 q^{82} +9.89949 q^{83} -4.00000 q^{85} -2.00000 q^{86} -2.00000 q^{88} -7.07107 q^{89} +4.00000 q^{92} -2.82843 q^{94} +20.0000 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{11} + 2 q^{16} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 2 q^{32} + 20 q^{37} + 4 q^{43} + 4 q^{44} - 8 q^{46} - 6 q^{50} + 4 q^{53} + 4 q^{58} + 2 q^{64} + 24 q^{67} + 24 q^{71} - 20 q^{74} - 8 q^{79} - 8 q^{85} - 4 q^{86} - 4 q^{88} + 8 q^{92} + 40 q^{95}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^11 + 2 * q^16 - 4 * q^22 + 8 * q^23 + 6 * q^25 - 4 * q^29 - 2 * q^32 + 20 * q^37 + 4 * q^43 + 4 * q^44 - 8 * q^46 - 6 * q^50 + 4 * q^53 + 4 * q^58 + 2 * q^64 + 24 * q^67 + 24 * q^71 - 20 * q^74 - 8 * q^79 - 8 * q^85 - 4 * q^86 - 4 * q^88 + 8 * q^92 + 40 * q^95

Introduction

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