LMFDB - Embedded newform 2842.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, 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("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 406)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} +2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{29} -4.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -8.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -8.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} -2.00000 q^{45} -6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{50} -8.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -4.00000 q^{54} -8.00000 q^{55} +4.00000 q^{57} +1.00000 q^{58} +4.00000 q^{59} +4.00000 q^{60} -4.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +8.00000 q^{66} -4.00000 q^{67} +4.00000 q^{68} +8.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} -2.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} +4.00000 q^{78} -12.0000 q^{79} -2.00000 q^{80} -11.0000 q^{81} +8.00000 q^{82} -8.00000 q^{85} +8.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -4.00000 q^{89} +2.00000 q^{90} -4.00000 q^{93} +6.00000 q^{94} +4.00000 q^{95} +2.00000 q^{96} -4.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.00000	0.408248
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	4.00000	0.769800
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.00000	−0.730297
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−8.00000	−1.39262
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	−4.00000	−0.640513
$40$	2.00000	0.316228
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	4.00000	0.603023
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	1.00000	0.141421
$51$	−8.00000	−1.12022
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−4.00000	−0.544331
$55$	−8.00000	−1.07872
$56$	0	0
$57$	4.00000	0.529813
$58$	1.00000	0.131306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	4.00000	0.516398
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	8.00000	0.984732
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−2.00000	−0.232495
$75$	2.00000	0.230940
$76$	−2.00000	−0.229416
$77$	0	0
$78$	4.00000	0.452911
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	8.00000	0.883452
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	8.00000	0.862662
$87$	2.00000	0.214423
$88$	−4.00000	−0.426401
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	−4.00000	−0.414781
$94$	6.00000	0.618853
$95$	4.00000	0.410391
$96$	2.00000	0.204124
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	−1.00000	−0.100000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	8.00000	0.792118
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	8.00000	0.762770
$111$	−4.00000	−0.379663
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	2.00000	0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−4.00000	−0.365148
$121$	5.00000	0.454545
$122$	4.00000	0.362143
$123$	16.0000	1.44267
$124$	2.00000	0.179605
$125$	12.0000	1.07331
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	16.0000	1.40872
$130$	4.00000	0.350823
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−8.00000	−0.696311
$133$	0	0
$134$	4.00000	0.345547
$135$	−8.00000	−0.688530
$136$	−4.00000	−0.342997
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−12.0000	−0.993127
$147$	0	0
$148$	2.00000	0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−2.00000	−0.163299
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	2.00000	0.162221
$153$	4.00000	0.323381
$154$	0	0
$155$	−4.00000	−0.321288
$156$	−4.00000	−0.320256
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	12.0000	0.954669
$159$	−12.0000	−0.951662
$160$	2.00000	0.158114
$161$	0	0
$162$	11.0000	0.864242
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−8.00000	−0.624695
$165$	16.0000	1.24560
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	8.00000	0.613572
$171$	−2.00000	−0.152944
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	−8.00000	−0.601317
$178$	4.00000	0.299813
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.00000	−0.149071
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−4.00000	−0.294086
$186$	4.00000	0.293294
$187$	16.0000	1.17004
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−4.00000	−0.290191
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−2.00000	−0.144338
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	4.00000	0.287183
$195$	8.00000	0.572892
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−4.00000	−0.284268
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	−12.0000	−0.844317
$203$	0	0
$204$	−8.00000	−0.560112
$205$	16.0000	1.11749
$206$	−16.0000	−1.11477
$207$	0	0
$208$	2.00000	0.138675
$209$	−8.00000	−0.553372
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000	0.412082
$213$	−16.0000	−1.09630
$214$	12.0000	0.820303
$215$	16.0000	1.09119
$216$	−4.00000	−0.272166
$217$	0	0
$218$	14.0000	0.948200
$219$	−24.0000	−1.62177
$220$	−8.00000	−0.539360
$221$	8.00000	0.538138
$222$	4.00000	0.268462
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−18.0000	−1.19734
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	4.00000	0.264906
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	1.00000	0.0656532
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−2.00000	−0.130744
$235$	12.0000	0.782794
$236$	4.00000	0.260378
$237$	24.0000	1.55897
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	4.00000	0.258199
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−5.00000	−0.321412
$243$	10.0000	0.641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−16.0000	−1.02012
$247$	−4.00000	−0.254514
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	0	0
$254$	−12.0000	−0.752947
$255$	16.0000	1.00196
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−16.0000	−0.996116
$259$	0	0
$260$	−4.00000	−0.248069
$261$	−1.00000	−0.0618984
$262$	2.00000	0.123560
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	8.00000	0.492366
$265$	−12.0000	−0.737154
$266$	0	0
$267$	8.00000	0.489592
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	8.00000	0.486864
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	−20.0000	−1.19952
$279$	2.00000	0.119737
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	−12.0000	−0.714590
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	8.00000	0.474713
$285$	−8.00000	−0.473879
$286$	−8.00000	−0.473050
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−2.00000	−0.117444
$291$	8.00000	0.468968
$292$	12.0000	0.702247
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	−2.00000	−0.116248
$297$	16.0000	0.928414
$298$	−18.0000	−1.04271
$299$	0	0
$300$	2.00000	0.115470
$301$	0	0
$302$	0	0
$303$	−24.0000	−1.37876
$304$	−2.00000	−0.114708
$305$	8.00000	0.458079
$306$	−4.00000	−0.228665
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	0	0
$309$	−32.0000	−1.82042
$310$	4.00000	0.227185
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	4.00000	0.226455
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	−12.0000	−0.675053
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	12.0000	0.672927
$319$	−4.00000	−0.223957
$320$	−2.00000	−0.111803
$321$	24.0000	1.33955
$322$	0	0
$323$	−8.00000	−0.445132
$324$	−11.0000	−0.611111
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	28.0000	1.54840
$328$	8.00000	0.441726
$329$	0	0
$330$	−16.0000	−0.880771
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	−12.0000	−0.656611
$335$	8.00000	0.437087
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	9.00000	0.489535
$339$	−36.0000	−1.95525
$340$	−8.00000	−0.433861
$341$	8.00000	0.433224
$342$	2.00000	0.108148
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	−4.00000	−0.213201
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	8.00000	0.425195
$355$	−16.0000	−0.849192
$356$	−4.00000	−0.212000
$357$	0	0
$358$	4.00000	0.211407
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	22.0000	1.15629
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−24.0000	−1.25622
$366$	−8.00000	−0.418167
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	4.00000	0.207950
$371$	0	0
$372$	−4.00000	−0.207390
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	−16.0000	−0.827340
$375$	−24.0000	−1.23935
$376$	6.00000	0.309426
$377$	−2.00000	−0.103005
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	4.00000	0.205196
$381$	−24.0000	−1.22956
$382$	−20.0000	−1.02329
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−8.00000	−0.406663
$388$	−4.00000	−0.203069
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	−8.00000	−0.405096
$391$	0	0
$392$	0	0
$393$	4.00000	0.201773
$394$	−2.00000	−0.100759
$395$	24.0000	1.20757
$396$	4.00000	0.201008
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−8.00000	−0.399004
$403$	4.00000	0.199254
$404$	12.0000	0.597022
$405$	22.0000	1.09319
$406$	0	0
$407$	8.00000	0.396545
$408$	8.00000	0.396059
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	−16.0000	−0.790184
$411$	20.0000	0.986527
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−40.0000	−1.95881
$418$	8.00000	0.391293
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−20.0000	−0.973585
$423$	−6.00000	−0.291730
$424$	−6.00000	−0.291386
$425$	−4.00000	−0.194029
$426$	16.0000	0.775203
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−16.0000	−0.772487
$430$	−16.0000	−0.771589
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	4.00000	0.192450
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−14.0000	−0.670478
$437$	0	0
$438$	24.0000	1.14676
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	8.00000	0.381385
$441$	0	0
$442$	−8.00000	−0.380521
$443$	32.0000	1.52037	0.760183	−	0.649709i	$-0.225109\pi$
$443$	32.0000	1.52037	0.760183	+	0.649709i	$0.225109\pi$
$444$	−4.00000	−0.189832
$445$	8.00000	0.379236
$446$	−4.00000	−0.189405
$447$	−36.0000	−1.70274
$448$	0	0
$449$	−38.0000	−1.79333	−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000	−1.79333	−0.896665	+	0.442709i	$0.854018\pi$
$450$	1.00000	0.0471405
$451$	−32.0000	−1.50682
$452$	18.0000	0.846649
$453$	0	0
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−4.00000	−0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	4.00000	0.186908
$459$	16.0000	0.746816
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−1.00000	−0.0464238
$465$	8.00000	0.370991
$466$	−10.0000	−0.463241
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	−12.0000	−0.553519
$471$	24.0000	1.10586
$472$	−4.00000	−0.184115
$473$	−32.0000	−1.47136
$474$	−24.0000	−1.10236
$475$	2.00000	0.0917663
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	−4.00000	−0.182574
$481$	4.00000	0.182384
$482$	22.0000	1.00207
$483$	0	0
$484$	5.00000	0.227273
$485$	8.00000	0.363261
$486$	−10.0000	−0.453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	4.00000	0.181071
$489$	−8.00000	−0.361773
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	16.0000	0.721336
$493$	−4.00000	−0.180151
$494$	4.00000	0.179969
$495$	−8.00000	−0.359573
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	12.0000	0.536656
$501$	−24.0000	−1.07224
$502$	−2.00000	−0.0892644
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	0	0
$507$	18.0000	0.799408
$508$	12.0000	0.532414
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	−16.0000	−0.708492
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	6.00000	0.264649
$515$	−32.0000	−1.41009
$516$	16.0000	0.704361
$517$	−24.0000	−1.05552
$518$	0	0
$519$	12.0000	0.526742
$520$	4.00000	0.175412
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	1.00000	0.0437688
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	−28.0000	−1.22086
$527$	8.00000	0.348485
$528$	−8.00000	−0.348155
$529$	−23.0000	−1.00000
$530$	12.0000	0.521247
$531$	4.00000	0.173585
$532$	0	0
$533$	−16.0000	−0.693037
$534$	−8.00000	−0.346194
$535$	24.0000	1.03761
$536$	4.00000	0.172774
$537$	8.00000	0.345225
$538$	0	0
$539$	0	0
$540$	−8.00000	−0.344265
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−2.00000	−0.0859074
$543$	44.0000	1.88822
$544$	−4.00000	−0.171499
$545$	28.0000	1.19939
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−10.0000	−0.427179
$549$	−4.00000	−0.170716
$550$	4.00000	0.170561
$551$	2.00000	0.0852029
$552$	0	0
$553$	0	0
$554$	6.00000	0.254916
$555$	8.00000	0.339581
$556$	20.0000	0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−2.00000	−0.0846668
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−32.0000	−1.35104
$562$	−22.0000	−0.928014
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	12.0000	0.505291
$565$	−36.0000	−1.51453
$566$	−20.0000	−0.840663
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	8.00000	0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	8.00000	0.334497
$573$	−40.0000	−1.67102
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	1.00000	0.0415945
$579$	−20.0000	−0.831172
$580$	2.00000	0.0830455
$581$	0	0
$582$	−8.00000	−0.331611
$583$	24.0000	0.993978
$584$	−12.0000	−0.496564
$585$	−4.00000	−0.165380
$586$	−24.0000	−0.991431
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	8.00000	0.329355
$591$	−4.00000	−0.164538
$592$	2.00000	0.0821995
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	−16.0000	−0.656488
$595$	0	0
$596$	18.0000	0.737309
$597$	16.0000	0.654836
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−2.00000	−0.0816497
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−10.0000	−0.406558
$606$	24.0000	0.974933
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−8.00000	−0.323911
$611$	−12.0000	−0.485468
$612$	4.00000	0.161690
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−34.0000	−1.37213
$615$	−32.0000	−1.29036
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	32.0000	1.28723
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−10.0000	−0.400963
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	16.0000	0.638978
$628$	−12.0000	−0.478852
$629$	8.00000	0.318981
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	12.0000	0.477334
$633$	−40.0000	−1.58986
$634$	−30.0000	−1.19145
$635$	−24.0000	−0.952411
$636$	−12.0000	−0.475831
$637$	0	0
$638$	4.00000	0.158362
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−24.0000	−0.947204
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−32.0000	−1.26000
$646$	8.00000	0.314756
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	11.0000	0.432121
$649$	16.0000	0.628055
$650$	2.00000	0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−28.0000	−1.09489
$655$	4.00000	0.156293
$656$	−8.00000	−0.312348
$657$	12.0000	0.468165
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	16.0000	0.622799
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−8.00000	−0.310929
$663$	−16.0000	−0.621389
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	12.0000	0.464294
$669$	−8.00000	−0.309298
$670$	−8.00000	−0.309067
$671$	−16.0000	−0.617673
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	26.0000	1.00148
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	−52.0000	−1.99852	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	−52.0000	−1.99852	−0.999261	+	0.0384331i	$0.987763\pi$
$678$	36.0000	1.38257
$679$	0	0
$680$	8.00000	0.306786
$681$	−56.0000	−2.14592
$682$	−8.00000	−0.306336
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−2.00000	−0.0764719
$685$	20.0000	0.764161
$686$	0	0
$687$	8.00000	0.305219
$688$	−8.00000	−0.304997
$689$	12.0000	0.457164
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	12.0000	0.455514
$695$	−40.0000	−1.51729
$696$	−2.00000	−0.0758098
$697$	−32.0000	−1.21209
$698$	26.0000	0.984115
$699$	−20.0000	−0.756469
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−8.00000	−0.301941
$703$	−4.00000	−0.150863
$704$	4.00000	0.150756
$705$	−24.0000	−0.903892
$706$	−22.0000	−0.827981
$707$	0	0
$708$	−8.00000	−0.300658
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	16.0000	0.600469
$711$	−12.0000	−0.450035
$712$	4.00000	0.149906
$713$	0	0
$714$	0	0
$715$	−16.0000	−0.598366
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−4.00000	−0.149279
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	15.0000	0.558242
$723$	44.0000	1.63638
$724$	−22.0000	−0.817624
$725$	1.00000	0.0371391
$726$	10.0000	0.371135
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	24.0000	0.888280
$731$	−32.0000	−1.18356
$732$	8.00000	0.295689
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	8.00000	0.294484
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	−4.00000	−0.147043
$741$	8.00000	0.293887
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	4.00000	0.146647
$745$	−36.0000	−1.31894
$746$	−34.0000	−1.24483
$747$	0	0
$748$	16.0000	0.585018
$749$	0	0
$750$	24.0000	0.876356
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−6.00000	−0.218797
$753$	−4.00000	−0.145768
$754$	2.00000	0.0728357
$755$	0	0
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	24.0000	0.869428
$763$	0	0
$764$	20.0000	0.723575
$765$	−8.00000	−0.289241
$766$	36.0000	1.30073
$767$	8.00000	0.288863
$768$	−2.00000	−0.0721688
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	10.0000	0.359908
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	8.00000	0.287554
$775$	−2.00000	−0.0718421
$776$	4.00000	0.143592
$777$	0	0
$778$	−26.0000	−0.932145
$779$	16.0000	0.573259
$780$	8.00000	0.286446
$781$	32.0000	1.14505
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	24.0000	0.856597
$786$	−4.00000	−0.142675
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	2.00000	0.0712470
$789$	−56.0000	−1.99365
$790$	−24.0000	−0.853882
$791$	0	0
$792$	−4.00000	−0.142134
$793$	−8.00000	−0.284088
$794$	14.0000	0.496841
$795$	24.0000	0.851192
$796$	−8.00000	−0.283552
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	−4.00000	−0.141333
$802$	−30.0000	−1.05934
$803$	48.0000	1.69388
$804$	8.00000	0.282138
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−12.0000	−0.422159
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−22.0000	−0.773001
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	−8.00000	−0.280400
$815$	−8.00000	−0.280228
$816$	−8.00000	−0.280056
$817$	16.0000	0.559769
$818$	12.0000	0.419570
$819$	0	0
$820$	16.0000	0.558744
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	−20.0000	−0.697580
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−16.0000	−0.557386
$825$	8.00000	0.278524
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	2.00000	0.0693375
$833$	0	0
$834$	40.0000	1.38509
$835$	−24.0000	−0.830554
$836$	−8.00000	−0.276686
$837$	8.00000	0.276520
$838$	−4.00000	−0.138178
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−22.0000	−0.758170
$843$	−44.0000	−1.51544
$844$	20.0000	0.688428
$845$	18.0000	0.619219
$846$	6.00000	0.206284
$847$	0	0
$848$	6.00000	0.206041
$849$	−40.0000	−1.37280
$850$	4.00000	0.137199
$851$	0	0
$852$	−16.0000	−0.548151
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	12.0000	0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	16.0000	0.546231
$859$	46.0000	1.56950	0.784750	−	0.619813i	$-0.212791\pi$
$859$	46.0000	1.56950	0.784750	+	0.619813i	$0.212791\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	40.0000	1.36241
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−4.00000	−0.136083
$865$	12.0000	0.408012
$866$	24.0000	0.815553
$867$	2.00000	0.0679236
$868$	0	0
$869$	−48.0000	−1.62829
$870$	4.00000	0.135613
$871$	−8.00000	−0.271070
$872$	14.0000	0.474100
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	−24.0000	−0.810885
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	32.0000	1.07995
$879$	−48.0000	−1.61900
$880$	−8.00000	−0.269680
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	8.00000	0.269069
$885$	16.0000	0.537834
$886$	−32.0000	−1.07506
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	−8.00000	−0.268161
$891$	−44.0000	−1.47406
$892$	4.00000	0.133930
$893$	12.0000	0.401565
$894$	36.0000	1.20402
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	38.0000	1.26808
$899$	−2.00000	−0.0667037
$900$	−1.00000	−0.0333333
$901$	24.0000	0.799556
$902$	32.0000	1.06548
$903$	0	0
$904$	−18.0000	−0.598671
$905$	44.0000	1.46261
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	28.0000	0.929213
$909$	12.0000	0.398015
$910$	0	0
$911$	52.0000	1.72284	0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000	1.72284	0.861418	+	0.507896i	$0.169577\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−10.0000	−0.330771
$915$	−16.0000	−0.528944
$916$	−4.00000	−0.132164
$917$	0	0
$918$	−16.0000	−0.528079
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	−68.0000	−2.24068
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	16.0000	0.525793
$927$	16.0000	0.525509
$928$	1.00000	0.0328266
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	10.0000	0.327561
$933$	−20.0000	−0.654771
$934$	−22.0000	−0.719862
$935$	−32.0000	−1.04651
$936$	−2.00000	−0.0653720
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	12.0000	0.391397
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	−24.0000	−0.781962
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	32.0000	1.04041
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	24.0000	0.779484
$949$	24.0000	0.779073
$950$	−2.00000	−0.0648886
$951$	−60.0000	−1.94563
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−6.00000	−0.194257
$955$	−40.0000	−1.29437
$956$	0	0
$957$	8.00000	0.258603
$958$	14.0000	0.452319
$959$	0	0
$960$	4.00000	0.129099
$961$	−27.0000	−0.870968
$962$	−4.00000	−0.128965
$963$	−12.0000	−0.386695
$964$	−22.0000	−0.708572
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	−5.00000	−0.160706
$969$	16.0000	0.513994
$970$	−8.00000	−0.256865
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	8.00000	0.256337
$975$	4.00000	0.128103
$976$	−4.00000	−0.128037
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	8.00000	0.255812
$979$	−16.0000	−0.511362
$980$	0	0
$981$	−14.0000	−0.446986
$982$	−20.0000	−0.638226
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	−16.0000	−0.510061
$985$	−4.00000	−0.127451
$986$	4.00000	0.127386
$987$	0	0
$988$	−4.00000	−0.127257
$989$	0	0
$990$	8.00000	0.254257
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−2.00000	−0.0635001
$993$	−16.0000	−0.507745
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	−12.0000	−0.379853
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.a.1.1		1
7.6	odd	2		406.2.a.c.1.1	✓	1
21.20	even	2		3654.2.a.n.1.1		1
28.27	even	2		3248.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
406.2.a.c.1.1	✓	1	7.6	odd	2
2842.2.a.a.1.1		1	1.1	even	1	trivial
3248.2.a.c.1.1		1	28.27	even	2
3654.2.a.n.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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