LMFDB - Embedded newform 9702.2.a.dd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{8} +3.46410 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -3.46410 q^{17} +1.46410 q^{19} +3.46410 q^{20} -1.00000 q^{22} +6.92820 q^{23} +7.00000 q^{25} -2.00000 q^{26} +6.00000 q^{29} +1.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +8.92820 q^{37} +1.46410 q^{38} +3.46410 q^{40} -3.46410 q^{41} +2.92820 q^{43} -1.00000 q^{44} +6.92820 q^{46} +2.53590 q^{47} +7.00000 q^{50} -2.00000 q^{52} -12.9282 q^{53} -3.46410 q^{55} +6.00000 q^{58} -6.92820 q^{59} -2.00000 q^{61} +1.46410 q^{62} +1.00000 q^{64} -6.92820 q^{65} +1.07180 q^{67} -3.46410 q^{68} +12.0000 q^{71} +7.46410 q^{73} +8.92820 q^{74} +1.46410 q^{76} +2.92820 q^{79} +3.46410 q^{80} -3.46410 q^{82} -16.3923 q^{83} -12.0000 q^{85} +2.92820 q^{86} -1.00000 q^{88} +12.9282 q^{89} +6.92820 q^{92} +2.53590 q^{94} +5.07180 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{11} - 4 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{22} + 14 q^{25} - 4 q^{26} + 12 q^{29} - 4 q^{31} + 2 q^{32} + 4 q^{37} - 4 q^{38} - 8 q^{43} - 2 q^{44} + 12 q^{47} + 14 q^{50} - 4 q^{52} - 12 q^{53} + 12 q^{58} - 4 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{67} + 24 q^{71} + 8 q^{73} + 4 q^{74} - 4 q^{76} - 8 q^{79} - 12 q^{83} - 24 q^{85} - 8 q^{86} - 2 q^{88} + 12 q^{89} + 12 q^{94} + 24 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 - 4 * q^13 + 2 * q^16 - 4 * q^19 - 2 * q^22 + 14 * q^25 - 4 * q^26 + 12 * q^29 - 4 * q^31 + 2 * q^32 + 4 * q^37 - 4 * q^38 - 8 * q^43 - 2 * q^44 + 12 * q^47 + 14 * q^50 - 4 * q^52 - 12 * q^53 + 12 * q^58 - 4 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^67 + 24 * q^71 + 8 * q^73 + 4 * q^74 - 4 * q^76 - 8 * q^79 - 12 * q^83 - 24 * q^85 - 8 * q^86 - 2 * q^88 + 12 * q^89 + 12 * q^94 + 24 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.46410	1.09545
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	3.46410	0.774597
$21$	0	0
$22$	−1.00000	−0.213201
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	0	0
$36$	0	0
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	1.46410	0.237509
$39$	0	0
$40$	3.46410	0.547723
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	2.92820	0.446547	0.223273	−	0.974756i	$-0.428326\pi$
$43$	2.92820	0.446547	0.223273	+	0.974756i	$0.428326\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	6.92820	1.02151
$47$	2.53590	0.369899	0.184949	−	0.982748i	$-0.440788\pi$
$47$	2.53590	0.369899	0.184949	+	0.982748i	$0.440788\pi$
$48$	0	0
$49$	0	0
$50$	7.00000	0.989949
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−12.9282	−1.77583	−0.887913	−	0.460012i	$-0.847845\pi$
$53$	−12.9282	−1.77583	−0.887913	+	0.460012i	$0.847845\pi$
$54$	0	0
$55$	−3.46410	−0.467099
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	1.46410	0.185941
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.92820	−0.859338
$66$	0	0
$67$	1.07180	0.130941	0.0654704	−	0.997855i	$-0.479145\pi$
$67$	1.07180	0.130941	0.0654704	+	0.997855i	$0.479145\pi$
$68$	−3.46410	−0.420084
$69$	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
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	8.92820	1.03788
$75$	0	0
$76$	1.46410	0.167944
$77$	0	0
$78$	0	0
$79$	2.92820	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$80$	3.46410	0.387298
$81$	0	0
$82$	−3.46410	−0.382546
$83$	−16.3923	−1.79929	−0.899645	−	0.436623i	$-0.856174\pi$
$83$	−16.3923	−1.79929	−0.899645	+	0.436623i	$0.856174\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	2.92820	0.315756
$87$	0	0
$88$	−1.00000	−0.106600
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	0	0
$92$	6.92820	0.722315
$93$	0	0
$94$	2.53590	0.261558
$95$	5.07180	0.520355
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	7.00000	0.700000
$101$	7.85641	0.781742	0.390871	−	0.920446i	$-0.372174\pi$
$101$	7.85641	0.781742	0.390871	+	0.920446i	$0.372174\pi$
$102$	0	0
$103$	6.53590	0.644001	0.322001	−	0.946739i	$-0.395645\pi$
$103$	6.53590	0.644001	0.322001	+	0.946739i	$0.395645\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−12.9282	−1.25570
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	−3.46410	−0.330289
$111$	0	0
$112$	0	0
$113$	19.8564	1.86793	0.933967	−	0.357360i	$-0.116323\pi$
$113$	19.8564	1.86793	0.933967	+	0.357360i	$0.116323\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	6.00000	0.557086
$117$	0	0
$118$	−6.92820	−0.637793
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	0	0
$124$	1.46410	0.131480
$125$	6.92820	0.619677
$126$	0	0
$127$	2.92820	0.259836	0.129918	−	0.991525i	$-0.458529\pi$
$127$	2.92820	0.259836	0.129918	+	0.991525i	$0.458529\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.92820	−0.607644
$131$	16.3923	1.43220	0.716101	−	0.697997i	$-0.245925\pi$
$131$	16.3923	1.43220	0.716101	+	0.697997i	$0.245925\pi$
$132$	0	0
$133$	0	0
$134$	1.07180	0.0925891
$135$	0	0
$136$	−3.46410	−0.297044
$137$	19.8564	1.69645	0.848224	−	0.529638i	$-0.177672\pi$
$137$	19.8564	1.69645	0.848224	+	0.529638i	$0.177672\pi$
$138$	0	0
$139$	6.53590	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$139$	6.53590	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	2.00000	0.167248
$144$	0	0
$145$	20.7846	1.72607
$146$	7.46410	0.617733
$147$	0	0
$148$	8.92820	0.733894
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	1.46410	0.118754
$153$	0	0
$154$	0	0
$155$	5.07180	0.407377
$156$	0	0
$157$	−18.3923	−1.46787	−0.733933	−	0.679222i	$-0.762317\pi$
$157$	−18.3923	−1.46787	−0.733933	+	0.679222i	$0.762317\pi$
$158$	2.92820	0.232955
$159$	0	0
$160$	3.46410	0.273861
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−16.3923	−1.27229
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	0	0
$172$	2.92820	0.223273
$173$	12.9282	0.982913	0.491457	−	0.870902i	$-0.336465\pi$
$173$	12.9282	0.982913	0.491457	+	0.870902i	$0.336465\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	12.9282	0.969010
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	14.3923	1.06977	0.534886	−	0.844924i	$-0.320355\pi$
$181$	14.3923	1.06977	0.534886	+	0.844924i	$0.320355\pi$
$182$	0	0
$183$	0	0
$184$	6.92820	0.510754
$185$	30.9282	2.27389
$186$	0	0
$187$	3.46410	0.253320
$188$	2.53590	0.184949
$189$	0	0
$190$	5.07180	0.367947
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	20.3923	1.44557	0.722786	−	0.691072i	$-0.242861\pi$
$199$	20.3923	1.44557	0.722786	+	0.691072i	$0.242861\pi$
$200$	7.00000	0.494975
$201$	0	0
$202$	7.85641	0.552775
$203$	0	0
$204$	0	0
$205$	−12.0000	−0.838116
$206$	6.53590	0.455378
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−1.46410	−0.101274
$210$	0	0
$211$	−24.7846	−1.70624	−0.853121	−	0.521712i	$-0.825293\pi$
$211$	−24.7846	−1.70624	−0.853121	+	0.521712i	$0.825293\pi$
$212$	−12.9282	−0.887913
$213$	0	0
$214$	−6.92820	−0.473602
$215$	10.1436	0.691787
$216$	0	0
$217$	0	0
$218$	−11.8564	−0.803017
$219$	0	0
$220$	−3.46410	−0.233550
$221$	6.92820	0.466041
$222$	0	0
$223$	−12.3923	−0.829850	−0.414925	−	0.909856i	$-0.636192\pi$
$223$	−12.3923	−0.829850	−0.414925	+	0.909856i	$0.636192\pi$
$224$	0	0
$225$	0	0
$226$	19.8564	1.32083
$227$	16.3923	1.08800	0.543998	−	0.839087i	$-0.316910\pi$
$227$	16.3923	1.08800	0.543998	+	0.839087i	$0.316910\pi$
$228$	0	0
$229$	−13.3205	−0.880244	−0.440122	−	0.897938i	$-0.645065\pi$
$229$	−13.3205	−0.880244	−0.440122	+	0.897938i	$0.645065\pi$
$230$	24.0000	1.58251
$231$	0	0
$232$	6.00000	0.393919
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	0	0
$235$	8.78461	0.573045
$236$	−6.92820	−0.450988
$237$	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
$241$	−30.3923	−1.95774	−0.978870	−	0.204482i	$-0.934449\pi$
$241$	−30.3923	−1.95774	−0.978870	+	0.204482i	$0.934449\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	−2.92820	−0.186317
$248$	1.46410	0.0929705
$249$	0	0
$250$	6.92820	0.438178
$251$	17.0718	1.07756	0.538781	−	0.842446i	$-0.318885\pi$
$251$	17.0718	1.07756	0.538781	+	0.842446i	$0.318885\pi$
$252$	0	0
$253$	−6.92820	−0.435572
$254$	2.92820	0.183732
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.928203	−0.0578997	−0.0289499	−	0.999581i	$-0.509216\pi$
$257$	−0.928203	−0.0578997	−0.0289499	+	0.999581i	$0.509216\pi$
$258$	0	0
$259$	0	0
$260$	−6.92820	−0.429669
$261$	0	0
$262$	16.3923	1.01272
$263$	18.9282	1.16716	0.583582	−	0.812055i	$-0.301651\pi$
$263$	18.9282	1.16716	0.583582	+	0.812055i	$0.301651\pi$
$264$	0	0
$265$	−44.7846	−2.75110
$266$	0	0
$267$	0	0
$268$	1.07180	0.0654704
$269$	−24.2487	−1.47847	−0.739235	−	0.673448i	$-0.764813\pi$
$269$	−24.2487	−1.47847	−0.739235	+	0.673448i	$0.764813\pi$
$270$	0	0
$271$	−16.7846	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$271$	−16.7846	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	19.8564	1.19957
$275$	−7.00000	−0.422116
$276$	0	0
$277$	15.8564	0.952719	0.476360	−	0.879251i	$-0.341956\pi$
$277$	15.8564	0.952719	0.476360	+	0.879251i	$0.341956\pi$
$278$	6.53590	0.391997
$279$	0	0
$280$	0	0
$281$	−19.8564	−1.18453	−0.592267	−	0.805742i	$-0.701767\pi$
$281$	−19.8564	−1.18453	−0.592267	+	0.805742i	$0.701767\pi$
$282$	0	0
$283$	6.53590	0.388519	0.194259	−	0.980950i	$-0.437770\pi$
$283$	6.53590	0.388519	0.194259	+	0.980950i	$0.437770\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	−5.00000	−0.294118
$290$	20.7846	1.22051
$291$	0	0
$292$	7.46410	0.436804
$293$	−11.0718	−0.646821	−0.323411	−	0.946259i	$-0.604830\pi$
$293$	−11.0718	−0.646821	−0.323411	+	0.946259i	$0.604830\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	8.92820	0.518941
$297$	0	0
$298$	6.00000	0.347571
$299$	−13.8564	−0.801337
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	0	0
$304$	1.46410	0.0839720
$305$	−6.92820	−0.396708
$306$	0	0
$307$	−17.4641	−0.996729	−0.498364	−	0.866968i	$-0.666066\pi$
$307$	−17.4641	−0.996729	−0.498364	+	0.866968i	$0.666066\pi$
$308$	0	0
$309$	0	0
$310$	5.07180	0.288059
$311$	7.60770	0.431393	0.215696	−	0.976460i	$-0.430798\pi$
$311$	7.60770	0.431393	0.215696	+	0.976460i	$0.430798\pi$
$312$	0	0
$313$	3.07180	0.173628	0.0868141	−	0.996225i	$-0.472331\pi$
$313$	3.07180	0.173628	0.0868141	+	0.996225i	$0.472331\pi$
$314$	−18.3923	−1.03794
$315$	0	0
$316$	2.92820	0.164724
$317$	0.928203	0.0521331	0.0260665	−	0.999660i	$-0.491702\pi$
$317$	0.928203	0.0521331	0.0260665	+	0.999660i	$0.491702\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	3.46410	0.193649
$321$	0	0
$322$	0	0
$323$	−5.07180	−0.282202
$324$	0	0
$325$	−14.0000	−0.776580
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−3.46410	−0.191273
$329$	0	0
$330$	0	0
$331$	−22.9282	−1.26025	−0.630124	−	0.776495i	$-0.716996\pi$
$331$	−22.9282	−1.26025	−0.630124	+	0.776495i	$0.716996\pi$
$332$	−16.3923	−0.899645
$333$	0	0
$334$	5.07180	0.277516
$335$	3.71281	0.202853
$336$	0	0
$337$	7.07180	0.385225	0.192613	−	0.981275i	$-0.438304\pi$
$337$	7.07180	0.385225	0.192613	+	0.981275i	$0.438304\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−12.0000	−0.650791
$341$	−1.46410	−0.0792855
$342$	0	0
$343$	0	0
$344$	2.92820	0.157878
$345$	0	0
$346$	12.9282	0.695025
$347$	−20.7846	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$348$	0	0
$349$	30.7846	1.64786	0.823931	−	0.566690i	$-0.191776\pi$
$349$	30.7846	1.64786	0.823931	+	0.566690i	$0.191776\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	41.5692	2.20627
$356$	12.9282	0.685193
$357$	0	0
$358$	−20.7846	−1.09850
$359$	18.9282	0.998992	0.499496	−	0.866316i	$-0.333518\pi$
$359$	18.9282	0.998992	0.499496	+	0.866316i	$0.333518\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	14.3923	0.756443
$363$	0	0
$364$	0	0
$365$	25.8564	1.35339
$366$	0	0
$367$	15.3205	0.799724	0.399862	−	0.916575i	$-0.369058\pi$
$367$	15.3205	0.799724	0.399862	+	0.916575i	$0.369058\pi$
$368$	6.92820	0.361158
$369$	0	0
$370$	30.9282	1.60788
$371$	0	0
$372$	0	0
$373$	−25.7128	−1.33136	−0.665679	−	0.746238i	$-0.731858\pi$
$373$	−25.7128	−1.33136	−0.665679	+	0.746238i	$0.731858\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	2.53590	0.130779
$377$	−12.0000	−0.618031
$378$	0	0
$379$	9.85641	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$379$	9.85641	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$380$	5.07180	0.260178
$381$	0	0
$382$	20.7846	1.06343
$383$	21.4641	1.09676	0.548382	−	0.836228i	$-0.315244\pi$
$383$	21.4641	1.09676	0.548382	+	0.836228i	$0.315244\pi$
$384$	0	0
$385$	0	0
$386$	26.0000	1.32337
$387$	0	0
$388$	−2.00000	−0.101535
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	−18.0000	−0.906827
$395$	10.1436	0.510380
$396$	0	0
$397$	−18.3923	−0.923083	−0.461542	−	0.887119i	$-0.652704\pi$
$397$	−18.3923	−0.923083	−0.461542	+	0.887119i	$0.652704\pi$
$398$	20.3923	1.02217
$399$	0	0
$400$	7.00000	0.350000
$401$	−31.8564	−1.59083	−0.795417	−	0.606063i	$-0.792748\pi$
$401$	−31.8564	−1.59083	−0.795417	+	0.606063i	$0.792748\pi$
$402$	0	0
$403$	−2.92820	−0.145864
$404$	7.85641	0.390871
$405$	0	0
$406$	0	0
$407$	−8.92820	−0.442555
$408$	0	0
$409$	−25.3205	−1.25202	−0.626009	−	0.779816i	$-0.715313\pi$
$409$	−25.3205	−1.25202	−0.626009	+	0.779816i	$0.715313\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	6.53590	0.322001
$413$	0	0
$414$	0	0
$415$	−56.7846	−2.78745
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−1.46410	−0.0716116
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	17.7128	0.863270	0.431635	−	0.902048i	$-0.357937\pi$
$421$	17.7128	0.863270	0.431635	+	0.902048i	$0.357937\pi$
$422$	−24.7846	−1.20650
$423$	0	0
$424$	−12.9282	−0.627849
$425$	−24.2487	−1.17624
$426$	0	0
$427$	0	0
$428$	−6.92820	−0.334887
$429$	0	0
$430$	10.1436	0.489168
$431$	5.07180	0.244300	0.122150	−	0.992512i	$-0.461021\pi$
$431$	5.07180	0.244300	0.122150	+	0.992512i	$0.461021\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	−11.8564	−0.567819
$437$	10.1436	0.485234
$438$	0	0
$439$	−16.7846	−0.801086	−0.400543	−	0.916278i	$-0.631178\pi$
$439$	−16.7846	−0.801086	−0.400543	+	0.916278i	$0.631178\pi$
$440$	−3.46410	−0.165145
$441$	0	0
$442$	6.92820	0.329541
$443$	−20.7846	−0.987507	−0.493753	−	0.869602i	$-0.664375\pi$
$443$	−20.7846	−0.987507	−0.493753	+	0.869602i	$0.664375\pi$
$444$	0	0
$445$	44.7846	2.12299
$446$	−12.3923	−0.586793
$447$	0	0
$448$	0	0
$449$	−7.85641	−0.370767	−0.185383	−	0.982666i	$-0.559353\pi$
$449$	−7.85641	−0.370767	−0.185383	+	0.982666i	$0.559353\pi$
$450$	0	0
$451$	3.46410	0.163118
$452$	19.8564	0.933967
$453$	0	0
$454$	16.3923	0.769329
$455$	0	0
$456$	0	0
$457$	−11.8564	−0.554619	−0.277310	−	0.960781i	$-0.589443\pi$
$457$	−11.8564	−0.554619	−0.277310	+	0.960781i	$0.589443\pi$
$458$	−13.3205	−0.622426
$459$	0	0
$460$	24.0000	1.11901
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−19.8564	−0.919830
$467$	39.7128	1.83769	0.918845	−	0.394619i	$-0.129123\pi$
$467$	39.7128	1.83769	0.918845	+	0.394619i	$0.129123\pi$
$468$	0	0
$469$	0	0
$470$	8.78461	0.405204
$471$	0	0
$472$	−6.92820	−0.318896
$473$	−2.92820	−0.134639
$474$	0	0
$475$	10.2487	0.470243
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−17.8564	−0.814182
$482$	−30.3923	−1.38433
$483$	0	0
$484$	1.00000	0.0454545
$485$	−6.92820	−0.314594
$486$	0	0
$487$	−0.784610	−0.0355541	−0.0177770	−	0.999842i	$-0.505659\pi$
$487$	−0.784610	−0.0355541	−0.0177770	+	0.999842i	$0.505659\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	−25.8564	−1.16688	−0.583442	−	0.812155i	$-0.698294\pi$
$491$	−25.8564	−1.16688	−0.583442	+	0.812155i	$0.698294\pi$
$492$	0	0
$493$	−20.7846	−0.936092
$494$	−2.92820	−0.131746
$495$	0	0
$496$	1.46410	0.0657401
$497$	0	0
$498$	0	0
$499$	−22.9282	−1.02641	−0.513204	−	0.858267i	$-0.671541\pi$
$499$	−22.9282	−1.02641	−0.513204	+	0.858267i	$0.671541\pi$
$500$	6.92820	0.309839
$501$	0	0
$502$	17.0718	0.761952
$503$	5.07180	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$503$	5.07180	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$504$	0	0
$505$	27.2154	1.21107
$506$	−6.92820	−0.307996
$507$	0	0
$508$	2.92820	0.129918
$509$	−6.67949	−0.296063	−0.148032	−	0.988983i	$-0.547294\pi$
$509$	−6.67949	−0.296063	−0.148032	+	0.988983i	$0.547294\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−0.928203	−0.0409413
$515$	22.6410	0.997682
$516$	0	0
$517$	−2.53590	−0.111529
$518$	0	0
$519$	0	0
$520$	−6.92820	−0.303822
$521$	23.0718	1.01079	0.505397	−	0.862887i	$-0.331346\pi$
$521$	23.0718	1.01079	0.505397	+	0.862887i	$0.331346\pi$
$522$	0	0
$523$	−3.60770	−0.157753	−0.0788767	−	0.996884i	$-0.525133\pi$
$523$	−3.60770	−0.157753	−0.0788767	+	0.996884i	$0.525133\pi$
$524$	16.3923	0.716101
$525$	0	0
$526$	18.9282	0.825309
$527$	−5.07180	−0.220931
$528$	0	0
$529$	25.0000	1.08696
$530$	−44.7846	−1.94532
$531$	0	0
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	−24.0000	−1.03761
$536$	1.07180	0.0462946
$537$	0	0
$538$	−24.2487	−1.04544
$539$	0	0
$540$	0	0
$541$	29.7128	1.27745	0.638727	−	0.769434i	$-0.279461\pi$
$541$	29.7128	1.27745	0.638727	+	0.769434i	$0.279461\pi$
$542$	−16.7846	−0.720961
$543$	0	0
$544$	−3.46410	−0.148522
$545$	−41.0718	−1.75932
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	19.8564	0.848224
$549$	0	0
$550$	−7.00000	−0.298481
$551$	8.78461	0.374237
$552$	0	0
$553$	0	0
$554$	15.8564	0.673674
$555$	0	0
$556$	6.53590	0.277184
$557$	33.7128	1.42846	0.714229	−	0.699912i	$-0.246778\pi$
$557$	33.7128	1.42846	0.714229	+	0.699912i	$0.246778\pi$
$558$	0	0
$559$	−5.85641	−0.247700
$560$	0	0
$561$	0	0
$562$	−19.8564	−0.837592
$563$	30.2487	1.27483	0.637416	−	0.770520i	$-0.280003\pi$
$563$	30.2487	1.27483	0.637416	+	0.770520i	$0.280003\pi$
$564$	0	0
$565$	68.7846	2.89379
$566$	6.53590	0.274724
$567$	0	0
$568$	12.0000	0.503509
$569$	4.14359	0.173708	0.0868542	−	0.996221i	$-0.472319\pi$
$569$	4.14359	0.173708	0.0868542	+	0.996221i	$0.472319\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0	0
$575$	48.4974	2.02248
$576$	0	0
$577$	−10.7846	−0.448969	−0.224485	−	0.974478i	$-0.572070\pi$
$577$	−10.7846	−0.448969	−0.224485	+	0.974478i	$0.572070\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	20.7846	0.863034
$581$	0	0
$582$	0	0
$583$	12.9282	0.535431
$584$	7.46410	0.308867
$585$	0	0
$586$	−11.0718	−0.457372
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	2.14359	0.0883252
$590$	−24.0000	−0.988064
$591$	0	0
$592$	8.92820	0.366947
$593$	−27.4641	−1.12782	−0.563908	−	0.825838i	$-0.690703\pi$
$593$	−27.4641	−1.12782	−0.563908	+	0.825838i	$0.690703\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	−13.8564	−0.566631
$599$	25.8564	1.05646	0.528232	−	0.849100i	$-0.322855\pi$
$599$	25.8564	1.05646	0.528232	+	0.849100i	$0.322855\pi$
$600$	0	0
$601$	2.39230	0.0975842	0.0487921	−	0.998809i	$-0.484463\pi$
$601$	2.39230	0.0975842	0.0487921	+	0.998809i	$0.484463\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	3.46410	0.140836
$606$	0	0
$607$	−35.7128	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$607$	−35.7128	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$608$	1.46410	0.0593772
$609$	0	0
$610$	−6.92820	−0.280515
$611$	−5.07180	−0.205183
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−17.4641	−0.704794
$615$	0	0
$616$	0	0
$617$	−7.85641	−0.316287	−0.158144	−	0.987416i	$-0.550551\pi$
$617$	−7.85641	−0.316287	−0.158144	+	0.987416i	$0.550551\pi$
$618$	0	0
$619$	−23.7128	−0.953098	−0.476549	−	0.879148i	$-0.658113\pi$
$619$	−23.7128	−0.953098	−0.476549	+	0.879148i	$0.658113\pi$
$620$	5.07180	0.203688
$621$	0	0
$622$	7.60770	0.305041
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	3.07180	0.122774
$627$	0	0
$628$	−18.3923	−0.733933
$629$	−30.9282	−1.23319
$630$	0	0
$631$	40.7846	1.62361	0.811805	−	0.583929i	$-0.198485\pi$
$631$	40.7846	1.62361	0.811805	+	0.583929i	$0.198485\pi$
$632$	2.92820	0.116478
$633$	0	0
$634$	0.928203	0.0368637
$635$	10.1436	0.402536
$636$	0	0
$637$	0	0
$638$	−6.00000	−0.237542
$639$	0	0
$640$	3.46410	0.136931
$641$	−4.14359	−0.163662	−0.0818311	−	0.996646i	$-0.526077\pi$
$641$	−4.14359	−0.163662	−0.0818311	+	0.996646i	$0.526077\pi$
$642$	0	0
$643$	−1.07180	−0.0422675	−0.0211338	−	0.999777i	$-0.506728\pi$
$643$	−1.07180	−0.0422675	−0.0211338	+	0.999777i	$0.506728\pi$
$644$	0	0
$645$	0	0
$646$	−5.07180	−0.199547
$647$	35.3205	1.38859	0.694296	−	0.719689i	$-0.255716\pi$
$647$	35.3205	1.38859	0.694296	+	0.719689i	$0.255716\pi$
$648$	0	0
$649$	6.92820	0.271956
$650$	−14.0000	−0.549125
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−21.7128	−0.849688	−0.424844	−	0.905267i	$-0.639671\pi$
$653$	−21.7128	−0.849688	−0.424844	+	0.905267i	$0.639671\pi$
$654$	0	0
$655$	56.7846	2.21876
$656$	−3.46410	−0.135250
$657$	0	0
$658$	0	0
$659$	30.9282	1.20479	0.602396	−	0.798197i	$-0.294213\pi$
$659$	30.9282	1.20479	0.602396	+	0.798197i	$0.294213\pi$
$660$	0	0
$661$	4.24871	0.165256	0.0826279	−	0.996580i	$-0.473669\pi$
$661$	4.24871	0.165256	0.0826279	+	0.996580i	$0.473669\pi$
$662$	−22.9282	−0.891130
$663$	0	0
$664$	−16.3923	−0.636145
$665$	0	0
$666$	0	0
$667$	41.5692	1.60957
$668$	5.07180	0.196234
$669$	0	0
$670$	3.71281	0.143438
$671$	2.00000	0.0772091
$672$	0	0
$673$	7.07180	0.272598	0.136299	−	0.990668i	$-0.456479\pi$
$673$	7.07180	0.272598	0.136299	+	0.990668i	$0.456479\pi$
$674$	7.07180	0.272395
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−38.7846	−1.49061	−0.745307	−	0.666722i	$-0.767697\pi$
$677$	−38.7846	−1.49061	−0.745307	+	0.666722i	$0.767697\pi$
$678$	0	0
$679$	0	0
$680$	−12.0000	−0.460179
$681$	0	0
$682$	−1.46410	−0.0560633
$683$	−44.7846	−1.71364	−0.856818	−	0.515619i	$-0.827562\pi$
$683$	−44.7846	−1.71364	−0.856818	+	0.515619i	$0.827562\pi$
$684$	0	0
$685$	68.7846	2.62812
$686$	0	0
$687$	0	0
$688$	2.92820	0.111637
$689$	25.8564	0.985051
$690$	0	0
$691$	46.9282	1.78523	0.892616	−	0.450817i	$-0.148867\pi$
$691$	46.9282	1.78523	0.892616	+	0.450817i	$0.148867\pi$
$692$	12.9282	0.491457
$693$	0	0
$694$	−20.7846	−0.788973
$695$	22.6410	0.858823
$696$	0	0
$697$	12.0000	0.454532
$698$	30.7846	1.16521
$699$	0	0
$700$	0	0
$701$	9.71281	0.366848	0.183424	−	0.983034i	$-0.441282\pi$
$701$	9.71281	0.366848	0.183424	+	0.983034i	$0.441282\pi$
$702$	0	0
$703$	13.0718	0.493012
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−0.928203	−0.0349334
$707$	0	0
$708$	0	0
$709$	5.21539	0.195868	0.0979340	−	0.995193i	$-0.468777\pi$
$709$	5.21539	0.195868	0.0979340	+	0.995193i	$0.468777\pi$
$710$	41.5692	1.56007
$711$	0	0
$712$	12.9282	0.484505
$713$	10.1436	0.379881
$714$	0	0
$715$	6.92820	0.259100
$716$	−20.7846	−0.776757
$717$	0	0
$718$	18.9282	0.706394
$719$	25.1769	0.938940	0.469470	−	0.882948i	$-0.344445\pi$
$719$	25.1769	0.938940	0.469470	+	0.882948i	$0.344445\pi$
$720$	0	0
$721$	0	0
$722$	−16.8564	−0.627330
$723$	0	0
$724$	14.3923	0.534886
$725$	42.0000	1.55984
$726$	0	0
$727$	29.1769	1.08211	0.541056	−	0.840987i	$-0.318025\pi$
$727$	29.1769	1.08211	0.541056	+	0.840987i	$0.318025\pi$
$728$	0	0
$729$	0	0
$730$	25.8564	0.956989
$731$	−10.1436	−0.375174
$732$	0	0
$733$	−39.8564	−1.47213	−0.736065	−	0.676911i	$-0.763318\pi$
$733$	−39.8564	−1.47213	−0.736065	+	0.676911i	$0.763318\pi$
$734$	15.3205	0.565490
$735$	0	0
$736$	6.92820	0.255377
$737$	−1.07180	−0.0394801
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	30.9282	1.13694
$741$	0	0
$742$	0	0
$743$	−41.5692	−1.52503	−0.762513	−	0.646972i	$-0.776035\pi$
$743$	−41.5692	−1.52503	−0.762513	+	0.646972i	$0.776035\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	−25.7128	−0.941413
$747$	0	0
$748$	3.46410	0.126660
$749$	0	0
$750$	0	0
$751$	16.7846	0.612479	0.306240	−	0.951954i	$-0.400929\pi$
$751$	16.7846	0.612479	0.306240	+	0.951954i	$0.400929\pi$
$752$	2.53590	0.0924747
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−55.4256	−2.01715
$756$	0	0
$757$	−18.7846	−0.682738	−0.341369	−	0.939929i	$-0.610891\pi$
$757$	−18.7846	−0.682738	−0.341369	+	0.939929i	$0.610891\pi$
$758$	9.85641	0.358001
$759$	0	0
$760$	5.07180	0.183973
$761$	−46.3923	−1.68172	−0.840860	−	0.541253i	$-0.817950\pi$
$761$	−46.3923	−1.68172	−0.840860	+	0.541253i	$0.817950\pi$
$762$	0	0
$763$	0	0
$764$	20.7846	0.751961
$765$	0	0
$766$	21.4641	0.775530
$767$	13.8564	0.500326
$768$	0	0
$769$	−35.4641	−1.27887	−0.639434	−	0.768846i	$-0.720831\pi$
$769$	−35.4641	−1.27887	−0.639434	+	0.768846i	$0.720831\pi$
$770$	0	0
$771$	0	0
$772$	26.0000	0.935760
$773$	−39.4641	−1.41943	−0.709713	−	0.704491i	$-0.751175\pi$
$773$	−39.4641	−1.41943	−0.709713	+	0.704491i	$0.751175\pi$
$774$	0	0
$775$	10.2487	0.368145
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	30.0000	1.07555
$779$	−5.07180	−0.181716
$780$	0	0
$781$	−12.0000	−0.429394
$782$	−24.0000	−0.858238
$783$	0	0
$784$	0	0
$785$	−63.7128	−2.27401
$786$	0	0
$787$	29.1769	1.04004	0.520022	−	0.854153i	$-0.325924\pi$
$787$	29.1769	1.04004	0.520022	+	0.854153i	$0.325924\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	10.1436	0.360893
$791$	0	0
$792$	0	0
$793$	4.00000	0.142044
$794$	−18.3923	−0.652718
$795$	0	0
$796$	20.3923	0.722786
$797$	−39.4641	−1.39789	−0.698945	−	0.715175i	$-0.746347\pi$
$797$	−39.4641	−1.39789	−0.698945	+	0.715175i	$0.746347\pi$
$798$	0	0
$799$	−8.78461	−0.310777
$800$	7.00000	0.247487
$801$	0	0
$802$	−31.8564	−1.12489
$803$	−7.46410	−0.263402
$804$	0	0
$805$	0	0
$806$	−2.92820	−0.103142
$807$	0	0
$808$	7.85641	0.276387
$809$	−24.9282	−0.876429	−0.438214	−	0.898870i	$-0.644389\pi$
$809$	−24.9282	−0.876429	−0.438214	+	0.898870i	$0.644389\pi$
$810$	0	0
$811$	−45.1769	−1.58638	−0.793188	−	0.608977i	$-0.791580\pi$
$811$	−45.1769	−1.58638	−0.793188	+	0.608977i	$0.791580\pi$
$812$	0	0
$813$	0	0
$814$	−8.92820	−0.312933
$815$	−13.8564	−0.485369
$816$	0	0
$817$	4.28719	0.149990
$818$	−25.3205	−0.885311
$819$	0	0
$820$	−12.0000	−0.419058
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−0.784610	−0.0273498	−0.0136749	−	0.999906i	$-0.504353\pi$
$823$	−0.784610	−0.0273498	−0.0136749	+	0.999906i	$0.504353\pi$
$824$	6.53590	0.227689
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	4.24871	0.147564	0.0737819	−	0.997274i	$-0.476493\pi$
$829$	4.24871	0.147564	0.0737819	+	0.997274i	$0.476493\pi$
$830$	−56.7846	−1.97102
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	17.5692	0.608008
$836$	−1.46410	−0.0506370
$837$	0	0
$838$	−36.0000	−1.24360
$839$	−26.5359	−0.916121	−0.458060	−	0.888921i	$-0.651456\pi$
$839$	−26.5359	−0.916121	−0.458060	+	0.888921i	$0.651456\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	17.7128	0.610424
$843$	0	0
$844$	−24.7846	−0.853121
$845$	−31.1769	−1.07252
$846$	0	0
$847$	0	0
$848$	−12.9282	−0.443956
$849$	0	0
$850$	−24.2487	−0.831724
$851$	61.8564	2.12041
$852$	0	0
$853$	−5.71281	−0.195603	−0.0978015	−	0.995206i	$-0.531181\pi$
$853$	−5.71281	−0.195603	−0.0978015	+	0.995206i	$0.531181\pi$
$854$	0	0
$855$	0	0
$856$	−6.92820	−0.236801
$857$	29.3205	1.00157	0.500785	−	0.865572i	$-0.333045\pi$
$857$	29.3205	1.00157	0.500785	+	0.865572i	$0.333045\pi$
$858$	0	0
$859$	−11.2154	−0.382664	−0.191332	−	0.981525i	$-0.561281\pi$
$859$	−11.2154	−0.382664	−0.191332	+	0.981525i	$0.561281\pi$
$860$	10.1436	0.345894
$861$	0	0
$862$	5.07180	0.172746
$863$	−3.21539	−0.109453	−0.0547266	−	0.998501i	$-0.517429\pi$
$863$	−3.21539	−0.109453	−0.0547266	+	0.998501i	$0.517429\pi$
$864$	0	0
$865$	44.7846	1.52272
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	0	0
$869$	−2.92820	−0.0993325
$870$	0	0
$871$	−2.14359	−0.0726329
$872$	−11.8564	−0.401509
$873$	0	0
$874$	10.1436	0.343112
$875$	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	$0.783085\pi$
$878$	−16.7846	−0.566453
$879$	0	0
$880$	−3.46410	−0.116775
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−14.1436	−0.475970	−0.237985	−	0.971269i	$-0.576487\pi$
$883$	−14.1436	−0.475970	−0.237985	+	0.971269i	$0.576487\pi$
$884$	6.92820	0.233021
$885$	0	0
$886$	−20.7846	−0.698273
$887$	37.8564	1.27109	0.635547	−	0.772062i	$-0.280775\pi$
$887$	37.8564	1.27109	0.635547	+	0.772062i	$0.280775\pi$
$888$	0	0
$889$	0	0
$890$	44.7846	1.50118
$891$	0	0
$892$	−12.3923	−0.414925
$893$	3.71281	0.124245
$894$	0	0
$895$	−72.0000	−2.40669
$896$	0	0
$897$	0	0
$898$	−7.85641	−0.262172
$899$	8.78461	0.292983
$900$	0	0
$901$	44.7846	1.49199
$902$	3.46410	0.115342
$903$	0	0
$904$	19.8564	0.660414
$905$	49.8564	1.65728
$906$	0	0
$907$	1.07180	0.0355884	0.0177942	−	0.999842i	$-0.494336\pi$
$907$	1.07180	0.0355884	0.0177942	+	0.999842i	$0.494336\pi$
$908$	16.3923	0.543998
$909$	0	0
$910$	0	0
$911$	−48.4974	−1.60679	−0.803396	−	0.595446i	$-0.796976\pi$
$911$	−48.4974	−1.60679	−0.803396	+	0.595446i	$0.796976\pi$
$912$	0	0
$913$	16.3923	0.542506
$914$	−11.8564	−0.392175
$915$	0	0
$916$	−13.3205	−0.440122
$917$	0	0
$918$	0	0
$919$	−5.85641	−0.193185	−0.0965925	−	0.995324i	$-0.530794\pi$
$919$	−5.85641	−0.193185	−0.0965925	+	0.995324i	$0.530794\pi$
$920$	24.0000	0.791257
$921$	0	0
$922$	−6.00000	−0.197599
$923$	−24.0000	−0.789970
$924$	0	0
$925$	62.4974	2.05490
$926$	8.00000	0.262896
$927$	0	0
$928$	6.00000	0.196960
$929$	−28.6410	−0.939681	−0.469841	−	0.882751i	$-0.655689\pi$
$929$	−28.6410	−0.939681	−0.469841	+	0.882751i	$0.655689\pi$
$930$	0	0
$931$	0	0
$932$	−19.8564	−0.650418
$933$	0	0
$934$	39.7128	1.29944
$935$	12.0000	0.392442
$936$	0	0
$937$	−6.39230	−0.208827	−0.104414	−	0.994534i	$-0.533297\pi$
$937$	−6.39230	−0.208827	−0.104414	+	0.994534i	$0.533297\pi$
$938$	0	0
$939$	0	0
$940$	8.78461	0.286522
$941$	−24.9282	−0.812636	−0.406318	−	0.913732i	$-0.633188\pi$
$941$	−24.9282	−0.812636	−0.406318	+	0.913732i	$0.633188\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−6.92820	−0.225494
$945$	0	0
$946$	−2.92820	−0.0952041
$947$	−22.1436	−0.719570	−0.359785	−	0.933035i	$-0.617150\pi$
$947$	−22.1436	−0.719570	−0.359785	+	0.933035i	$0.617150\pi$
$948$	0	0
$949$	−14.9282	−0.484590
$950$	10.2487	0.332512
$951$	0	0
$952$	0	0
$953$	−14.7846	−0.478920	−0.239460	−	0.970906i	$-0.576970\pi$
$953$	−14.7846	−0.478920	−0.239460	+	0.970906i	$0.576970\pi$
$954$	0	0
$955$	72.0000	2.32987
$956$	24.0000	0.776215
$957$	0	0
$958$	13.8564	0.447680
$959$	0	0
$960$	0	0
$961$	−28.8564	−0.930852
$962$	−17.8564	−0.575714
$963$	0	0
$964$	−30.3923	−0.978870
$965$	90.0666	2.89935
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−6.92820	−0.222451
$971$	39.7128	1.27444	0.637222	−	0.770680i	$-0.280083\pi$
$971$	39.7128	1.27444	0.637222	+	0.770680i	$0.280083\pi$
$972$	0	0
$973$	0	0
$974$	−0.784610	−0.0251405
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−35.5692	−1.13796	−0.568980	−	0.822351i	$-0.692662\pi$
$977$	−35.5692	−1.13796	−0.568980	+	0.822351i	$0.692662\pi$
$978$	0	0
$979$	−12.9282	−0.413187
$980$	0	0
$981$	0	0
$982$	−25.8564	−0.825111
$983$	−11.3205	−0.361068	−0.180534	−	0.983569i	$-0.557783\pi$
$983$	−11.3205	−0.361068	−0.180534	+	0.983569i	$0.557783\pi$
$984$	0	0
$985$	−62.3538	−1.98676
$986$	−20.7846	−0.661917
$987$	0	0
$988$	−2.92820	−0.0931586
$989$	20.2872	0.645095
$990$	0	0
$991$	−29.8564	−0.948420	−0.474210	−	0.880412i	$-0.657266\pi$
$991$	−29.8564	−0.948420	−0.474210	+	0.880412i	$0.657266\pi$
$992$	1.46410	0.0464853
$993$	0	0
$994$	0	0
$995$	70.6410	2.23947
$996$	0	0
$997$	44.6410	1.41380	0.706898	−	0.707316i	$-0.250094\pi$
$997$	44.6410	1.41380	0.706898	+	0.707316i	$0.250094\pi$
$998$	−22.9282	−0.725780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dd.1.2		2
3.2	odd	2		3234.2.a.x.1.1		2
7.6	odd	2		1386.2.a.p.1.1		2
21.20	even	2		462.2.a.h.1.2	✓	2
84.83	odd	2		3696.2.a.bc.1.2		2
231.230	odd	2		5082.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.h.1.2	✓	2	21.20	even	2
1386.2.a.p.1.1		2	7.6	odd	2
3234.2.a.x.1.1		2	3.2	odd	2
3696.2.a.bc.1.2		2	84.83	odd	2
5082.2.a.bu.1.2		2	231.230	odd	2
9702.2.a.dd.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{8} +3.46410 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -3.46410 q^{17} +1.46410 q^{19} +3.46410 q^{20} -1.00000 q^{22} +6.92820 q^{23} +7.00000 q^{25} -2.00000 q^{26} +6.00000 q^{29} +1.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +8.92820 q^{37} +1.46410 q^{38} +3.46410 q^{40} -3.46410 q^{41} +2.92820 q^{43} -1.00000 q^{44} +6.92820 q^{46} +2.53590 q^{47} +7.00000 q^{50} -2.00000 q^{52} -12.9282 q^{53} -3.46410 q^{55} +6.00000 q^{58} -6.92820 q^{59} -2.00000 q^{61} +1.46410 q^{62} +1.00000 q^{64} -6.92820 q^{65} +1.07180 q^{67} -3.46410 q^{68} +12.0000 q^{71} +7.46410 q^{73} +8.92820 q^{74} +1.46410 q^{76} +2.92820 q^{79} +3.46410 q^{80} -3.46410 q^{82} -16.3923 q^{83} -12.0000 q^{85} +2.92820 q^{86} -1.00000 q^{88} +12.9282 q^{89} +6.92820 q^{92} +2.53590 q^{94} +5.07180 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} - 4 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{22} + 14 q^{25} - 4 q^{26} + 12 q^{29} - 4 q^{31} + 2 q^{32} + 4 q^{37} - 4 q^{38} - 8 q^{43} - 2 q^{44} + 12 q^{47} + 14 q^{50} - 4 q^{52} - 12 q^{53} + 12 q^{58} - 4 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{67} + 24 q^{71} + 8 q^{73} + 4 q^{74} - 4 q^{76} - 8 q^{79} - 12 q^{83} - 24 q^{85} - 8 q^{86} - 2 q^{88} + 12 q^{89} + 12 q^{94} + 24 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 - 4 * q^13 + 2 * q^16 - 4 * q^19 - 2 * q^22 + 14 * q^25 - 4 * q^26 + 12 * q^29 - 4 * q^31 + 2 * q^32 + 4 * q^37 - 4 * q^38 - 8 * q^43 - 2 * q^44 + 12 * q^47 + 14 * q^50 - 4 * q^52 - 12 * q^53 + 12 * q^58 - 4 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^67 + 24 * q^71 + 8 * q^73 + 4 * q^74 - 4 * q^76 - 8 * q^79 - 12 * q^83 - 24 * q^85 - 8 * q^86 - 2 * q^88 + 12 * q^89 + 12 * q^94 + 24 * q^95 - 4 * q^97

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