LMFDB - Embedded newform 9702.2.a.dm.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(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.64575 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.64575 q^{5} +1.00000 q^{8} +2.64575 q^{10} -1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -5.29150 q^{19} +2.64575 q^{20} -1.00000 q^{22} +2.64575 q^{23} +2.00000 q^{25} +4.00000 q^{26} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +1.29150 q^{37} -5.29150 q^{38} +2.64575 q^{40} +9.00000 q^{41} +9.29150 q^{43} -1.00000 q^{44} +2.64575 q^{46} -3.93725 q^{47} +2.00000 q^{50} +4.00000 q^{52} -4.00000 q^{53} -2.64575 q^{55} -2.00000 q^{58} +6.58301 q^{59} -11.9373 q^{61} +4.00000 q^{62} +1.00000 q^{64} +10.5830 q^{65} +7.58301 q^{67} -3.00000 q^{68} +2.70850 q^{71} +15.2915 q^{73} +1.29150 q^{74} -5.29150 q^{76} +10.6458 q^{79} +2.64575 q^{80} +9.00000 q^{82} +15.5830 q^{83} -7.93725 q^{85} +9.29150 q^{86} -1.00000 q^{88} +2.70850 q^{89} +2.64575 q^{92} -3.93725 q^{94} -14.0000 q^{95} +11.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{22} + 4 q^{25} + 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{34} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 2 q^{44} + 8 q^{47} + 4 q^{50} + 8 q^{52} - 8 q^{53} - 4 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{67} - 6 q^{68} + 16 q^{71} + 20 q^{73} - 8 q^{74} + 16 q^{79} + 18 q^{82} + 10 q^{83} + 8 q^{86} - 2 q^{88} + 16 q^{89} + 8 q^{94} - 28 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 8 * q^13 + 2 * q^16 - 6 * q^17 - 2 * q^22 + 4 * q^25 + 8 * q^26 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^34 - 8 * q^37 + 18 * q^41 + 8 * q^43 - 2 * q^44 + 8 * q^47 + 4 * q^50 + 8 * q^52 - 8 * q^53 - 4 * q^58 - 8 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 - 6 * q^67 - 6 * q^68 + 16 * q^71 + 20 * q^73 - 8 * q^74 + 16 * q^79 + 18 * q^82 + 10 * q^83 + 8 * q^86 - 2 * q^88 + 16 * q^89 + 8 * q^94 - 28 * q^95 + 2 * 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
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	2.64575	0.836660
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	2.64575	0.591608
$21$	0	0
$22$	−1.00000	−0.213201
$23$	2.64575	0.551677	0.275839	−	0.961204i	$-0.411044\pi$
$23$	2.64575	0.551677	0.275839	+	0.961204i	$0.411044\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	4.00000	0.784465
$27$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	1.29150	0.212322	0.106161	−	0.994349i	$-0.466144\pi$
$37$	1.29150	0.212322	0.106161	+	0.994349i	$0.466144\pi$
$38$	−5.29150	−0.858395
$39$	0	0
$40$	2.64575	0.418330
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	9.29150	1.41694	0.708470	−	0.705740i	$-0.249386\pi$
$43$	9.29150	1.41694	0.708470	+	0.705740i	$0.249386\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	2.64575	0.390095
$47$	−3.93725	−0.574308	−0.287154	−	0.957885i	$-0.592709\pi$
$47$	−3.93725	−0.574308	−0.287154	+	0.957885i	$0.592709\pi$
$48$	0	0
$49$	0	0
$50$	2.00000	0.282843
$51$	0	0
$52$	4.00000	0.554700
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−2.64575	−0.356753
$56$	0	0
$57$	0	0
$58$	−2.00000	−0.262613
$59$	6.58301	0.857034	0.428517	−	0.903534i	$-0.359036\pi$
$59$	6.58301	0.857034	0.428517	+	0.903534i	$0.359036\pi$
$60$	0	0
$61$	−11.9373	−1.52841	−0.764204	−	0.644974i	$-0.776868\pi$
$61$	−11.9373	−1.52841	−0.764204	+	0.644974i	$0.776868\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	10.5830	1.31266
$66$	0	0
$67$	7.58301	0.926412	0.463206	−	0.886251i	$-0.346699\pi$
$67$	7.58301	0.926412	0.463206	+	0.886251i	$0.346699\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	0	0
$71$	2.70850	0.321440	0.160720	−	0.987000i	$-0.448618\pi$
$71$	2.70850	0.321440	0.160720	+	0.987000i	$0.448618\pi$
$72$	0	0
$73$	15.2915	1.78974	0.894868	−	0.446332i	$-0.147270\pi$
$73$	15.2915	1.78974	0.894868	+	0.446332i	$0.147270\pi$
$74$	1.29150	0.150134
$75$	0	0
$76$	−5.29150	−0.606977
$77$	0	0
$78$	0	0
$79$	10.6458	1.19774	0.598870	−	0.800846i	$-0.295616\pi$
$79$	10.6458	1.19774	0.598870	+	0.800846i	$0.295616\pi$
$80$	2.64575	0.295804
$81$	0	0
$82$	9.00000	0.993884
$83$	15.5830	1.71046	0.855229	−	0.518251i	$-0.173417\pi$
$83$	15.5830	1.71046	0.855229	+	0.518251i	$0.173417\pi$
$84$	0	0
$85$	−7.93725	−0.860916
$86$	9.29150	1.00193
$87$	0	0
$88$	−1.00000	−0.106600
$89$	2.70850	0.287100	0.143550	−	0.989643i	$-0.454148\pi$
$89$	2.70850	0.287100	0.143550	+	0.989643i	$0.454148\pi$
$90$	0	0
$91$	0	0
$92$	2.64575	0.275839
$93$	0	0
$94$	−3.93725	−0.406097
$95$	−14.0000	−1.43637
$96$	0	0
$97$	11.5830	1.17608	0.588038	−	0.808833i	$-0.299901\pi$
$97$	11.5830	1.17608	0.588038	+	0.808833i	$0.299901\pi$
$98$	0	0
$99$	0	0
$100$	2.00000	0.200000
$101$	7.29150	0.725532	0.362766	−	0.931880i	$-0.381832\pi$
$101$	7.29150	0.725532	0.362766	+	0.931880i	$0.381832\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	0	0
$109$	−3.93725	−0.377121	−0.188560	−	0.982062i	$-0.560382\pi$
$109$	−3.93725	−0.377121	−0.188560	+	0.982062i	$0.560382\pi$
$110$	−2.64575	−0.252262
$111$	0	0
$112$	0	0
$113$	−12.5830	−1.18371	−0.591855	−	0.806045i	$-0.701604\pi$
$113$	−12.5830	−1.18371	−0.591855	+	0.806045i	$0.701604\pi$
$114$	0	0
$115$	7.00000	0.652753
$116$	−2.00000	−0.185695
$117$	0	0
$118$	6.58301	0.606015
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.9373	−1.08075
$123$	0	0
$124$	4.00000	0.359211
$125$	−7.93725	−0.709930
$126$	0	0
$127$	2.64575	0.234772	0.117386	−	0.993086i	$-0.462548\pi$
$127$	2.64575	0.234772	0.117386	+	0.993086i	$0.462548\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.5830	0.928191
$131$	−18.5830	−1.62360	−0.811802	−	0.583932i	$-0.801513\pi$
$131$	−18.5830	−1.62360	−0.811802	+	0.583932i	$0.801513\pi$
$132$	0	0
$133$	0	0
$134$	7.58301	0.655072
$135$	0	0
$136$	−3.00000	−0.257248
$137$	13.2915	1.13557	0.567785	−	0.823177i	$-0.307801\pi$
$137$	13.2915	1.13557	0.567785	+	0.823177i	$0.307801\pi$
$138$	0	0
$139$	12.5830	1.06728	0.533638	−	0.845713i	$-0.320824\pi$
$139$	12.5830	1.06728	0.533638	+	0.845713i	$0.320824\pi$
$140$	0	0
$141$	0	0
$142$	2.70850	0.227292
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−5.29150	−0.439435
$146$	15.2915	1.26553
$147$	0	0
$148$	1.29150	0.106161
$149$	−19.8745	−1.62818	−0.814092	−	0.580737i	$-0.802765\pi$
$149$	−19.8745	−1.62818	−0.814092	+	0.580737i	$0.802765\pi$
$150$	0	0
$151$	22.6458	1.84289	0.921443	−	0.388515i	$-0.127012\pi$
$151$	22.6458	1.84289	0.921443	+	0.388515i	$0.127012\pi$
$152$	−5.29150	−0.429198
$153$	0	0
$154$	0	0
$155$	10.5830	0.850047
$156$	0	0
$157$	0.708497	0.0565442	0.0282721	−	0.999600i	$-0.491000\pi$
$157$	0.708497	0.0565442	0.0282721	+	0.999600i	$0.491000\pi$
$158$	10.6458	0.846931
$159$	0	0
$160$	2.64575	0.209165
$161$	0	0
$162$	0	0
$163$	−5.58301	−0.437295	−0.218647	−	0.975804i	$-0.570164\pi$
$163$	−5.58301	−0.437295	−0.218647	+	0.975804i	$0.570164\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	15.5830	1.20948
$167$	8.70850	0.673884	0.336942	−	0.941525i	$-0.390607\pi$
$167$	8.70850	0.673884	0.336942	+	0.941525i	$0.390607\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−7.93725	−0.608760
$171$	0	0
$172$	9.29150	0.708470
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	2.70850	0.203010
$179$	−4.70850	−0.351930	−0.175965	−	0.984396i	$-0.556304\pi$
$179$	−4.70850	−0.351930	−0.175965	+	0.984396i	$0.556304\pi$
$180$	0	0
$181$	5.29150	0.393314	0.196657	−	0.980472i	$-0.436991\pi$
$181$	5.29150	0.393314	0.196657	+	0.980472i	$0.436991\pi$
$182$	0	0
$183$	0	0
$184$	2.64575	0.195047
$185$	3.41699	0.251222
$186$	0	0
$187$	3.00000	0.219382
$188$	−3.93725	−0.287154
$189$	0	0
$190$	−14.0000	−1.01567
$191$	−6.70850	−0.485410	−0.242705	−	0.970100i	$-0.578035\pi$
$191$	−6.70850	−0.485410	−0.242705	+	0.970100i	$0.578035\pi$
$192$	0	0
$193$	−15.8745	−1.14267	−0.571336	−	0.820716i	$-0.693575\pi$
$193$	−15.8745	−1.14267	−0.571336	+	0.820716i	$0.693575\pi$
$194$	11.5830	0.831611
$195$	0	0
$196$	0	0
$197$	−21.8745	−1.55849	−0.779247	−	0.626717i	$-0.784398\pi$
$197$	−21.8745	−1.55849	−0.779247	+	0.626717i	$0.784398\pi$
$198$	0	0
$199$	19.1660	1.35864	0.679321	−	0.733841i	$-0.262274\pi$
$199$	19.1660	1.35864	0.679321	+	0.733841i	$0.262274\pi$
$200$	2.00000	0.141421
$201$	0	0
$202$	7.29150	0.513028
$203$	0	0
$204$	0	0
$205$	23.8118	1.66309
$206$	10.0000	0.696733
$207$	0	0
$208$	4.00000	0.277350
$209$	5.29150	0.366021
$210$	0	0
$211$	21.2915	1.46577	0.732884	−	0.680354i	$-0.238174\pi$
$211$	21.2915	1.46577	0.732884	+	0.680354i	$0.238174\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−9.00000	−0.615227
$215$	24.5830	1.67655
$216$	0	0
$217$	0	0
$218$	−3.93725	−0.266664
$219$	0	0
$220$	−2.64575	−0.178377
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−28.4575	−1.90566	−0.952828	−	0.303511i	$-0.901841\pi$
$223$	−28.4575	−1.90566	−0.952828	+	0.303511i	$0.901841\pi$
$224$	0	0
$225$	0	0
$226$	−12.5830	−0.837009
$227$	2.41699	0.160422	0.0802108	−	0.996778i	$-0.474441\pi$
$227$	2.41699	0.160422	0.0802108	+	0.996778i	$0.474441\pi$
$228$	0	0
$229$	−6.70850	−0.443310	−0.221655	−	0.975125i	$-0.571146\pi$
$229$	−6.70850	−0.443310	−0.221655	+	0.975125i	$0.571146\pi$
$230$	7.00000	0.461566
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−20.1660	−1.32112	−0.660560	−	0.750774i	$-0.729681\pi$
$233$	−20.1660	−1.32112	−0.660560	+	0.750774i	$0.729681\pi$
$234$	0	0
$235$	−10.4170	−0.679530
$236$	6.58301	0.428517
$237$	0	0
$238$	0	0
$239$	−2.70850	−0.175198	−0.0875991	−	0.996156i	$-0.527919\pi$
$239$	−2.70850	−0.175198	−0.0875991	+	0.996156i	$0.527919\pi$
$240$	0	0
$241$	27.1660	1.74992	0.874958	−	0.484198i	$-0.160889\pi$
$241$	27.1660	1.74992	0.874958	+	0.484198i	$0.160889\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−11.9373	−0.764204
$245$	0	0
$246$	0	0
$247$	−21.1660	−1.34676
$248$	4.00000	0.254000
$249$	0	0
$250$	−7.93725	−0.501996
$251$	23.2915	1.47015	0.735073	−	0.677988i	$-0.237148\pi$
$251$	23.2915	1.47015	0.735073	+	0.677988i	$0.237148\pi$
$252$	0	0
$253$	−2.64575	−0.166337
$254$	2.64575	0.166009
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.29150	0.579588	0.289794	−	0.957089i	$-0.406413\pi$
$257$	9.29150	0.579588	0.289794	+	0.957089i	$0.406413\pi$
$258$	0	0
$259$	0	0
$260$	10.5830	0.656330
$261$	0	0
$262$	−18.5830	−1.14806
$263$	−21.8745	−1.34884	−0.674420	−	0.738348i	$-0.735606\pi$
$263$	−21.8745	−1.34884	−0.674420	+	0.738348i	$0.735606\pi$
$264$	0	0
$265$	−10.5830	−0.650109
$266$	0	0
$267$	0	0
$268$	7.58301	0.463206
$269$	−10.6458	−0.649083	−0.324541	−	0.945871i	$-0.605210\pi$
$269$	−10.6458	−0.649083	−0.324541	+	0.945871i	$0.605210\pi$
$270$	0	0
$271$	−9.29150	−0.564419	−0.282209	−	0.959353i	$-0.591067\pi$
$271$	−9.29150	−0.564419	−0.282209	+	0.959353i	$0.591067\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	13.2915	0.802969
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−17.1660	−1.03141	−0.515703	−	0.856768i	$-0.672469\pi$
$277$	−17.1660	−1.03141	−0.515703	+	0.856768i	$0.672469\pi$
$278$	12.5830	0.754679
$279$	0	0
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	−30.4575	−1.81051	−0.905256	−	0.424867i	$-0.860321\pi$
$283$	−30.4575	−1.81051	−0.905256	+	0.424867i	$0.860321\pi$
$284$	2.70850	0.160720
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−5.29150	−0.310728
$291$	0	0
$292$	15.2915	0.894868
$293$	−14.5830	−0.851948	−0.425974	−	0.904735i	$-0.640069\pi$
$293$	−14.5830	−0.851948	−0.425974	+	0.904735i	$0.640069\pi$
$294$	0	0
$295$	17.4170	1.01406
$296$	1.29150	0.0750671
$297$	0	0
$298$	−19.8745	−1.15130
$299$	10.5830	0.612031
$300$	0	0
$301$	0	0
$302$	22.6458	1.30312
$303$	0	0
$304$	−5.29150	−0.303488
$305$	−31.5830	−1.80844
$306$	0	0
$307$	11.4170	0.651602	0.325801	−	0.945438i	$-0.394366\pi$
$307$	11.4170	0.651602	0.325801	+	0.945438i	$0.394366\pi$
$308$	0	0
$309$	0	0
$310$	10.5830	0.601074
$311$	22.5203	1.27701	0.638503	−	0.769619i	$-0.279554\pi$
$311$	22.5203	1.27701	0.638503	+	0.769619i	$0.279554\pi$
$312$	0	0
$313$	20.5830	1.16342	0.581710	−	0.813396i	$-0.302384\pi$
$313$	20.5830	1.16342	0.581710	+	0.813396i	$0.302384\pi$
$314$	0.708497	0.0399828
$315$	0	0
$316$	10.6458	0.598870
$317$	−9.22876	−0.518339	−0.259169	−	0.965832i	$-0.583449\pi$
$317$	−9.22876	−0.518339	−0.259169	+	0.965832i	$0.583449\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	2.64575	0.147902
$321$	0	0
$322$	0	0
$323$	15.8745	0.883281
$324$	0	0
$325$	8.00000	0.443760
$326$	−5.58301	−0.309214
$327$	0	0
$328$	9.00000	0.496942
$329$	0	0
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	15.5830	0.855229
$333$	0	0
$334$	8.70850	0.476508
$335$	20.0627	1.09614
$336$	0	0
$337$	6.70850	0.365435	0.182718	−	0.983165i	$-0.441511\pi$
$337$	6.70850	0.365435	0.182718	+	0.983165i	$0.441511\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	−7.93725	−0.430458
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	9.29150	0.500964
$345$	0	0
$346$	4.00000	0.215041
$347$	−29.5830	−1.58810	−0.794049	−	0.607853i	$-0.792031\pi$
$347$	−29.5830	−1.58810	−0.794049	+	0.607853i	$0.792031\pi$
$348$	0	0
$349$	13.2288	0.708119	0.354060	−	0.935223i	$-0.384801\pi$
$349$	13.2288	0.708119	0.354060	+	0.935223i	$0.384801\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−23.2915	−1.23968	−0.619841	−	0.784728i	$-0.712803\pi$
$353$	−23.2915	−1.23968	−0.619841	+	0.784728i	$0.712803\pi$
$354$	0	0
$355$	7.16601	0.380332
$356$	2.70850	0.143550
$357$	0	0
$358$	−4.70850	−0.248852
$359$	−20.5830	−1.08633	−0.543165	−	0.839626i	$-0.682774\pi$
$359$	−20.5830	−1.08633	−0.543165	+	0.839626i	$0.682774\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	5.29150	0.278115
$363$	0	0
$364$	0	0
$365$	40.4575	2.11764
$366$	0	0
$367$	−10.7085	−0.558979	−0.279490	−	0.960149i	$-0.590165\pi$
$367$	−10.7085	−0.558979	−0.279490	+	0.960149i	$0.590165\pi$
$368$	2.64575	0.137919
$369$	0	0
$370$	3.41699	0.177641
$371$	0	0
$372$	0	0
$373$	−15.8118	−0.818702	−0.409351	−	0.912377i	$-0.634245\pi$
$373$	−15.8118	−0.818702	−0.409351	+	0.912377i	$0.634245\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	−3.93725	−0.203048
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−27.5830	−1.41684	−0.708422	−	0.705789i	$-0.750593\pi$
$379$	−27.5830	−1.41684	−0.708422	+	0.705789i	$0.750593\pi$
$380$	−14.0000	−0.718185
$381$	0	0
$382$	−6.70850	−0.343237
$383$	18.7085	0.955960	0.477980	−	0.878371i	$-0.341369\pi$
$383$	18.7085	0.955960	0.477980	+	0.878371i	$0.341369\pi$
$384$	0	0
$385$	0	0
$386$	−15.8745	−0.807991
$387$	0	0
$388$	11.5830	0.588038
$389$	−31.9373	−1.61928	−0.809642	−	0.586925i	$-0.800338\pi$
$389$	−31.9373	−1.61928	−0.809642	+	0.586925i	$0.800338\pi$
$390$	0	0
$391$	−7.93725	−0.401404
$392$	0	0
$393$	0	0
$394$	−21.8745	−1.10202
$395$	28.1660	1.41719
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	19.1660	0.960705
$399$	0	0
$400$	2.00000	0.100000
$401$	35.8745	1.79149	0.895744	−	0.444571i	$-0.146644\pi$
$401$	35.8745	1.79149	0.895744	+	0.444571i	$0.146644\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	7.29150	0.362766
$405$	0	0
$406$	0	0
$407$	−1.29150	−0.0640174
$408$	0	0
$409$	−35.8745	−1.77388	−0.886940	−	0.461884i	$-0.847173\pi$
$409$	−35.8745	−1.77388	−0.886940	+	0.461884i	$0.847173\pi$
$410$	23.8118	1.17598
$411$	0	0
$412$	10.0000	0.492665
$413$	0	0
$414$	0	0
$415$	41.2288	2.02384
$416$	4.00000	0.196116
$417$	0	0
$418$	5.29150	0.258816
$419$	−9.87451	−0.482401	−0.241201	−	0.970475i	$-0.577541\pi$
$419$	−9.87451	−0.482401	−0.241201	+	0.970475i	$0.577541\pi$
$420$	0	0
$421$	6.12549	0.298538	0.149269	−	0.988797i	$-0.452308\pi$
$421$	6.12549	0.298538	0.149269	+	0.988797i	$0.452308\pi$
$422$	21.2915	1.03645
$423$	0	0
$424$	−4.00000	−0.194257
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	−9.00000	−0.435031
$429$	0	0
$430$	24.5830	1.18550
$431$	17.2915	0.832902	0.416451	−	0.909158i	$-0.363274\pi$
$431$	17.2915	0.832902	0.416451	+	0.909158i	$0.363274\pi$
$432$	0	0
$433$	−1.58301	−0.0760744	−0.0380372	−	0.999276i	$-0.512111\pi$
$433$	−1.58301	−0.0760744	−0.0380372	+	0.999276i	$0.512111\pi$
$434$	0	0
$435$	0	0
$436$	−3.93725	−0.188560
$437$	−14.0000	−0.669711
$438$	0	0
$439$	9.35425	0.446454	0.223227	−	0.974766i	$-0.428341\pi$
$439$	9.35425	0.446454	0.223227	+	0.974766i	$0.428341\pi$
$440$	−2.64575	−0.126131
$441$	0	0
$442$	−12.0000	−0.570782
$443$	31.1660	1.48074	0.740371	−	0.672199i	$-0.234650\pi$
$443$	31.1660	1.48074	0.740371	+	0.672199i	$0.234650\pi$
$444$	0	0
$445$	7.16601	0.339701
$446$	−28.4575	−1.34750
$447$	0	0
$448$	0	0
$449$	−30.4575	−1.43738	−0.718689	−	0.695331i	$-0.755258\pi$
$449$	−30.4575	−1.43738	−0.718689	+	0.695331i	$0.755258\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	−12.5830	−0.591855
$453$	0	0
$454$	2.41699	0.113435
$455$	0	0
$456$	0	0
$457$	−40.5830	−1.89839	−0.949196	−	0.314684i	$-0.898101\pi$
$457$	−40.5830	−1.89839	−0.949196	+	0.314684i	$0.898101\pi$
$458$	−6.70850	−0.313467
$459$	0	0
$460$	7.00000	0.326377
$461$	−24.5830	−1.14494	−0.572472	−	0.819924i	$-0.694016\pi$
$461$	−24.5830	−1.14494	−0.572472	+	0.819924i	$0.694016\pi$
$462$	0	0
$463$	39.0405	1.81437	0.907183	−	0.420735i	$-0.138228\pi$
$463$	39.0405	1.81437	0.907183	+	0.420735i	$0.138228\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−20.1660	−0.934172
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	0	0
$470$	−10.4170	−0.480500
$471$	0	0
$472$	6.58301	0.303007
$473$	−9.29150	−0.427224
$474$	0	0
$475$	−10.5830	−0.485582
$476$	0	0
$477$	0	0
$478$	−2.70850	−0.123884
$479$	−33.8745	−1.54777	−0.773883	−	0.633329i	$-0.781688\pi$
$479$	−33.8745	−1.54777	−0.773883	+	0.633329i	$0.781688\pi$
$480$	0	0
$481$	5.16601	0.235550
$482$	27.1660	1.23738
$483$	0	0
$484$	1.00000	0.0454545
$485$	30.6458	1.39155
$486$	0	0
$487$	21.8745	0.991229	0.495614	−	0.868543i	$-0.334943\pi$
$487$	21.8745	0.991229	0.495614	+	0.868543i	$0.334943\pi$
$488$	−11.9373	−0.540374
$489$	0	0
$490$	0	0
$491$	1.00000	0.0451294	0.0225647	−	0.999745i	$-0.492817\pi$
$491$	1.00000	0.0451294	0.0225647	+	0.999745i	$0.492817\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	−21.1660	−0.952304
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	−18.5830	−0.831890	−0.415945	−	0.909390i	$-0.636549\pi$
$499$	−18.5830	−0.831890	−0.415945	+	0.909390i	$0.636549\pi$
$500$	−7.93725	−0.354965
$501$	0	0
$502$	23.2915	1.03955
$503$	−1.29150	−0.0575853	−0.0287926	−	0.999585i	$-0.509166\pi$
$503$	−1.29150	−0.0575853	−0.0287926	+	0.999585i	$0.509166\pi$
$504$	0	0
$505$	19.2915	0.858461
$506$	−2.64575	−0.117618
$507$	0	0
$508$	2.64575	0.117386
$509$	25.1660	1.11546	0.557732	−	0.830021i	$-0.311672\pi$
$509$	25.1660	1.11546	0.557732	+	0.830021i	$0.311672\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.29150	0.409831
$515$	26.4575	1.16586
$516$	0	0
$517$	3.93725	0.173160
$518$	0	0
$519$	0	0
$520$	10.5830	0.464095
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	−2.70850	−0.118434	−0.0592172	−	0.998245i	$-0.518860\pi$
$523$	−2.70850	−0.118434	−0.0592172	+	0.998245i	$0.518860\pi$
$524$	−18.5830	−0.811802
$525$	0	0
$526$	−21.8745	−0.953774
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−16.0000	−0.695652
$530$	−10.5830	−0.459696
$531$	0	0
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	−23.8118	−1.02947
$536$	7.58301	0.327536
$537$	0	0
$538$	−10.6458	−0.458971
$539$	0	0
$540$	0	0
$541$	3.81176	0.163880	0.0819402	−	0.996637i	$-0.473888\pi$
$541$	3.81176	0.163880	0.0819402	+	0.996637i	$0.473888\pi$
$542$	−9.29150	−0.399104
$543$	0	0
$544$	−3.00000	−0.128624
$545$	−10.4170	−0.446215
$546$	0	0
$547$	14.7085	0.628890	0.314445	−	0.949276i	$-0.398182\pi$
$547$	14.7085	0.628890	0.314445	+	0.949276i	$0.398182\pi$
$548$	13.2915	0.567785
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	10.5830	0.450851
$552$	0	0
$553$	0	0
$554$	−17.1660	−0.729314
$555$	0	0
$556$	12.5830	0.533638
$557$	19.2915	0.817407	0.408704	−	0.912667i	$-0.365981\pi$
$557$	19.2915	0.817407	0.408704	+	0.912667i	$0.365981\pi$
$558$	0	0
$559$	37.1660	1.57195
$560$	0	0
$561$	0	0
$562$	3.00000	0.126547
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−33.2915	−1.40058
$566$	−30.4575	−1.28022
$567$	0	0
$568$	2.70850	0.113646
$569$	15.1660	0.635792	0.317896	−	0.948126i	$-0.397024\pi$
$569$	15.1660	0.635792	0.317896	+	0.948126i	$0.397024\pi$
$570$	0	0
$571$	−29.7490	−1.24496	−0.622479	−	0.782637i	$-0.713874\pi$
$571$	−29.7490	−1.24496	−0.622479	+	0.782637i	$0.713874\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	5.29150	0.220671
$576$	0	0
$577$	−22.7490	−0.947054	−0.473527	−	0.880779i	$-0.657019\pi$
$577$	−22.7490	−0.947054	−0.473527	+	0.880779i	$0.657019\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−5.29150	−0.219718
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	15.2915	0.632767
$585$	0	0
$586$	−14.5830	−0.602418
$587$	9.87451	0.407565	0.203782	−	0.979016i	$-0.434677\pi$
$587$	9.87451	0.407565	0.203782	+	0.979016i	$0.434677\pi$
$588$	0	0
$589$	−21.1660	−0.872130
$590$	17.4170	0.717046
$591$	0	0
$592$	1.29150	0.0530804
$593$	−25.7490	−1.05739	−0.528693	−	0.848813i	$-0.677318\pi$
$593$	−25.7490	−1.05739	−0.528693	+	0.848813i	$0.677318\pi$
$594$	0	0
$595$	0	0
$596$	−19.8745	−0.814092
$597$	0	0
$598$	10.5830	0.432771
$599$	−29.3542	−1.19938	−0.599691	−	0.800232i	$-0.704710\pi$
$599$	−29.3542	−1.19938	−0.599691	+	0.800232i	$0.704710\pi$
$600$	0	0
$601$	38.5830	1.57383	0.786917	−	0.617059i	$-0.211676\pi$
$601$	38.5830	1.57383	0.786917	+	0.617059i	$0.211676\pi$
$602$	0	0
$603$	0	0
$604$	22.6458	0.921443
$605$	2.64575	0.107565
$606$	0	0
$607$	13.1033	0.531845	0.265923	−	0.963994i	$-0.414323\pi$
$607$	13.1033	0.531845	0.265923	+	0.963994i	$0.414323\pi$
$608$	−5.29150	−0.214599
$609$	0	0
$610$	−31.5830	−1.27876
$611$	−15.7490	−0.637137
$612$	0	0
$613$	7.81176	0.315514	0.157757	−	0.987478i	$-0.449574\pi$
$613$	7.81176	0.315514	0.157757	+	0.987478i	$0.449574\pi$
$614$	11.4170	0.460752
$615$	0	0
$616$	0	0
$617$	14.7085	0.592142	0.296071	−	0.955166i	$-0.404324\pi$
$617$	14.7085	0.592142	0.296071	+	0.955166i	$0.404324\pi$
$618$	0	0
$619$	41.0000	1.64793	0.823965	−	0.566641i	$-0.191757\pi$
$619$	41.0000	1.64793	0.823965	+	0.566641i	$0.191757\pi$
$620$	10.5830	0.425024
$621$	0	0
$622$	22.5203	0.902980
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	20.5830	0.822662
$627$	0	0
$628$	0.708497	0.0282721
$629$	−3.87451	−0.154487
$630$	0	0
$631$	41.8745	1.66700	0.833499	−	0.552521i	$-0.186334\pi$
$631$	41.8745	1.66700	0.833499	+	0.552521i	$0.186334\pi$
$632$	10.6458	0.423465
$633$	0	0
$634$	−9.22876	−0.366521
$635$	7.00000	0.277787
$636$	0	0
$637$	0	0
$638$	2.00000	0.0791808
$639$	0	0
$640$	2.64575	0.104583
$641$	26.5830	1.04997	0.524983	−	0.851113i	$-0.324072\pi$
$641$	26.5830	1.04997	0.524983	+	0.851113i	$0.324072\pi$
$642$	0	0
$643$	1.16601	0.0459830	0.0229915	−	0.999736i	$-0.492681\pi$
$643$	1.16601	0.0459830	0.0229915	+	0.999736i	$0.492681\pi$
$644$	0	0
$645$	0	0
$646$	15.8745	0.624574
$647$	−26.3948	−1.03769	−0.518843	−	0.854870i	$-0.673637\pi$
$647$	−26.3948	−1.03769	−0.518843	+	0.854870i	$0.673637\pi$
$648$	0	0
$649$	−6.58301	−0.258406
$650$	8.00000	0.313786
$651$	0	0
$652$	−5.58301	−0.218647
$653$	45.1033	1.76503	0.882514	−	0.470287i	$-0.155850\pi$
$653$	45.1033	1.76503	0.882514	+	0.470287i	$0.155850\pi$
$654$	0	0
$655$	−49.1660	−1.92107
$656$	9.00000	0.351391
$657$	0	0
$658$	0	0
$659$	24.1660	0.941374	0.470687	−	0.882300i	$-0.344006\pi$
$659$	24.1660	0.941374	0.470687	+	0.882300i	$0.344006\pi$
$660$	0	0
$661$	−20.7085	−0.805467	−0.402734	−	0.915317i	$-0.631940\pi$
$661$	−20.7085	−0.805467	−0.402734	+	0.915317i	$0.631940\pi$
$662$	13.0000	0.505259
$663$	0	0
$664$	15.5830	0.604738
$665$	0	0
$666$	0	0
$667$	−5.29150	−0.204888
$668$	8.70850	0.336942
$669$	0	0
$670$	20.0627	0.775092
$671$	11.9373	0.460833
$672$	0	0
$673$	−19.4170	−0.748470	−0.374235	−	0.927334i	$-0.622095\pi$
$673$	−19.4170	−0.748470	−0.374235	+	0.927334i	$0.622095\pi$
$674$	6.70850	0.258402
$675$	0	0
$676$	3.00000	0.115385
$677$	−8.12549	−0.312288	−0.156144	−	0.987734i	$-0.549906\pi$
$677$	−8.12549	−0.312288	−0.156144	+	0.987734i	$0.549906\pi$
$678$	0	0
$679$	0	0
$680$	−7.93725	−0.304380
$681$	0	0
$682$	−4.00000	−0.153168
$683$	17.1660	0.656839	0.328420	−	0.944532i	$-0.393484\pi$
$683$	17.1660	0.656839	0.328420	+	0.944532i	$0.393484\pi$
$684$	0	0
$685$	35.1660	1.34362
$686$	0	0
$687$	0	0
$688$	9.29150	0.354235
$689$	−16.0000	−0.609551
$690$	0	0
$691$	−14.1660	−0.538900	−0.269450	−	0.963014i	$-0.586842\pi$
$691$	−14.1660	−0.538900	−0.269450	+	0.963014i	$0.586842\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	−29.5830	−1.12296
$695$	33.2915	1.26282
$696$	0	0
$697$	−27.0000	−1.02270
$698$	13.2288	0.500716
$699$	0	0
$700$	0	0
$701$	−42.4575	−1.60360	−0.801799	−	0.597594i	$-0.796124\pi$
$701$	−42.4575	−1.60360	−0.801799	+	0.597594i	$0.796124\pi$
$702$	0	0
$703$	−6.83399	−0.257749
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−23.2915	−0.876587
$707$	0	0
$708$	0	0
$709$	15.7490	0.591467	0.295733	−	0.955271i	$-0.404436\pi$
$709$	15.7490	0.591467	0.295733	+	0.955271i	$0.404436\pi$
$710$	7.16601	0.268936
$711$	0	0
$712$	2.70850	0.101505
$713$	10.5830	0.396337
$714$	0	0
$715$	−10.5830	−0.395782
$716$	−4.70850	−0.175965
$717$	0	0
$718$	−20.5830	−0.768151
$719$	−28.0627	−1.04656	−0.523282	−	0.852160i	$-0.675293\pi$
$719$	−28.0627	−1.04656	−0.523282	+	0.852160i	$0.675293\pi$
$720$	0	0
$721$	0	0
$722$	9.00000	0.334945
$723$	0	0
$724$	5.29150	0.196657
$725$	−4.00000	−0.148556
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	0	0
$730$	40.4575	1.49740
$731$	−27.8745	−1.03098
$732$	0	0
$733$	−11.9373	−0.440913	−0.220456	−	0.975397i	$-0.570755\pi$
$733$	−11.9373	−0.440913	−0.220456	+	0.975397i	$0.570755\pi$
$734$	−10.7085	−0.395258
$735$	0	0
$736$	2.64575	0.0975237
$737$	−7.58301	−0.279324
$738$	0	0
$739$	−24.5830	−0.904300	−0.452150	−	0.891942i	$-0.649343\pi$
$739$	−24.5830	−0.904300	−0.452150	+	0.891942i	$0.649343\pi$
$740$	3.41699	0.125611
$741$	0	0
$742$	0	0
$743$	18.7085	0.686348	0.343174	−	0.939272i	$-0.388498\pi$
$743$	18.7085	0.686348	0.343174	+	0.939272i	$0.388498\pi$
$744$	0	0
$745$	−52.5830	−1.92649
$746$	−15.8118	−0.578910
$747$	0	0
$748$	3.00000	0.109691
$749$	0	0
$750$	0	0
$751$	−29.2915	−1.06886	−0.534431	−	0.845212i	$-0.679474\pi$
$751$	−29.2915	−1.06886	−0.534431	+	0.845212i	$0.679474\pi$
$752$	−3.93725	−0.143577
$753$	0	0
$754$	−8.00000	−0.291343
$755$	59.9150	2.18053
$756$	0	0
$757$	−32.5830	−1.18425	−0.592125	−	0.805846i	$-0.701711\pi$
$757$	−32.5830	−1.18425	−0.592125	+	0.805846i	$0.701711\pi$
$758$	−27.5830	−1.00186
$759$	0	0
$760$	−14.0000	−0.507833
$761$	47.5830	1.72488	0.862441	−	0.506157i	$-0.168934\pi$
$761$	47.5830	1.72488	0.862441	+	0.506157i	$0.168934\pi$
$762$	0	0
$763$	0	0
$764$	−6.70850	−0.242705
$765$	0	0
$766$	18.7085	0.675965
$767$	26.3320	0.950794
$768$	0	0
$769$	30.7085	1.10738	0.553688	−	0.832724i	$-0.313220\pi$
$769$	30.7085	1.10738	0.553688	+	0.832724i	$0.313220\pi$
$770$	0	0
$771$	0	0
$772$	−15.8745	−0.571336
$773$	−8.06275	−0.289997	−0.144998	−	0.989432i	$-0.546318\pi$
$773$	−8.06275	−0.289997	−0.144998	+	0.989432i	$0.546318\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	11.5830	0.415806
$777$	0	0
$778$	−31.9373	−1.14501
$779$	−47.6235	−1.70629
$780$	0	0
$781$	−2.70850	−0.0969177
$782$	−7.93725	−0.283836
$783$	0	0
$784$	0	0
$785$	1.87451	0.0669041
$786$	0	0
$787$	−9.29150	−0.331206	−0.165603	−	0.986192i	$-0.552957\pi$
$787$	−9.29150	−0.331206	−0.165603	+	0.986192i	$0.552957\pi$
$788$	−21.8745	−0.779247
$789$	0	0
$790$	28.1660	1.00210
$791$	0	0
$792$	0	0
$793$	−47.7490	−1.69562
$794$	−6.00000	−0.212932
$795$	0	0
$796$	19.1660	0.679321
$797$	45.1033	1.59764	0.798820	−	0.601570i	$-0.205458\pi$
$797$	45.1033	1.59764	0.798820	+	0.601570i	$0.205458\pi$
$798$	0	0
$799$	11.8118	0.417870
$800$	2.00000	0.0707107
$801$	0	0
$802$	35.8745	1.26677
$803$	−15.2915	−0.539625
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	7.29150	0.256514
$809$	−45.3320	−1.59379	−0.796894	−	0.604119i	$-0.793525\pi$
$809$	−45.3320	−1.59379	−0.796894	+	0.604119i	$0.793525\pi$
$810$	0	0
$811$	−16.5830	−0.582308	−0.291154	−	0.956676i	$-0.594039\pi$
$811$	−16.5830	−0.582308	−0.291154	+	0.956676i	$0.594039\pi$
$812$	0	0
$813$	0	0
$814$	−1.29150	−0.0452671
$815$	−14.7712	−0.517414
$816$	0	0
$817$	−49.1660	−1.72010
$818$	−35.8745	−1.25432
$819$	0	0
$820$	23.8118	0.831543
$821$	30.3320	1.05859	0.529297	−	0.848436i	$-0.322456\pi$
$821$	30.3320	1.05859	0.529297	+	0.848436i	$0.322456\pi$
$822$	0	0
$823$	−12.1255	−0.422668	−0.211334	−	0.977414i	$-0.567781\pi$
$823$	−12.1255	−0.422668	−0.211334	+	0.977414i	$0.567781\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	0	0
$827$	−35.3320	−1.22861	−0.614307	−	0.789067i	$-0.710565\pi$
$827$	−35.3320	−1.22861	−0.614307	+	0.789067i	$0.710565\pi$
$828$	0	0
$829$	−19.8745	−0.690270	−0.345135	−	0.938553i	$-0.612167\pi$
$829$	−19.8745	−0.690270	−0.345135	+	0.938553i	$0.612167\pi$
$830$	41.2288	1.43107
$831$	0	0
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	23.0405	0.797350
$836$	5.29150	0.183010
$837$	0	0
$838$	−9.87451	−0.341109
$839$	−37.1033	−1.28095	−0.640473	−	0.767980i	$-0.721262\pi$
$839$	−37.1033	−1.28095	−0.640473	+	0.767980i	$0.721262\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.12549	0.211098
$843$	0	0
$844$	21.2915	0.732884
$845$	7.93725	0.273050
$846$	0	0
$847$	0	0
$848$	−4.00000	−0.137361
$849$	0	0
$850$	−6.00000	−0.205798
$851$	3.41699	0.117133
$852$	0	0
$853$	−14.5203	−0.497164	−0.248582	−	0.968611i	$-0.579965\pi$
$853$	−14.5203	−0.497164	−0.248582	+	0.968611i	$0.579965\pi$
$854$	0	0
$855$	0	0
$856$	−9.00000	−0.307614
$857$	9.83399	0.335923	0.167961	−	0.985794i	$-0.446282\pi$
$857$	9.83399	0.335923	0.167961	+	0.985794i	$0.446282\pi$
$858$	0	0
$859$	−48.7490	−1.66329	−0.831647	−	0.555304i	$-0.812602\pi$
$859$	−48.7490	−1.66329	−0.831647	+	0.555304i	$0.812602\pi$
$860$	24.5830	0.838274
$861$	0	0
$862$	17.2915	0.588951
$863$	33.1033	1.12685	0.563424	−	0.826168i	$-0.309484\pi$
$863$	33.1033	1.12685	0.563424	+	0.826168i	$0.309484\pi$
$864$	0	0
$865$	10.5830	0.359833
$866$	−1.58301	−0.0537927
$867$	0	0
$868$	0	0
$869$	−10.6458	−0.361132
$870$	0	0
$871$	30.3320	1.02776
$872$	−3.93725	−0.133332
$873$	0	0
$874$	−14.0000	−0.473557
$875$	0	0
$876$	0	0
$877$	10.7712	0.363719	0.181860	−	0.983325i	$-0.441788\pi$
$877$	10.7712	0.363719	0.181860	+	0.983325i	$0.441788\pi$
$878$	9.35425	0.315691
$879$	0	0
$880$	−2.64575	−0.0891883
$881$	23.1660	0.780483	0.390241	−	0.920713i	$-0.372392\pi$
$881$	23.1660	0.780483	0.390241	+	0.920713i	$0.372392\pi$
$882$	0	0
$883$	−34.4170	−1.15822	−0.579112	−	0.815248i	$-0.696601\pi$
$883$	−34.4170	−1.15822	−0.579112	+	0.815248i	$0.696601\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	31.1660	1.04704
$887$	−31.0405	−1.04224	−0.521119	−	0.853484i	$-0.674485\pi$
$887$	−31.0405	−1.04224	−0.521119	+	0.853484i	$0.674485\pi$
$888$	0	0
$889$	0	0
$890$	7.16601	0.240205
$891$	0	0
$892$	−28.4575	−0.952828
$893$	20.8340	0.697183
$894$	0	0
$895$	−12.4575	−0.416409
$896$	0	0
$897$	0	0
$898$	−30.4575	−1.01638
$899$	−8.00000	−0.266815
$900$	0	0
$901$	12.0000	0.399778
$902$	−9.00000	−0.299667
$903$	0	0
$904$	−12.5830	−0.418505
$905$	14.0000	0.465376
$906$	0	0
$907$	−15.8340	−0.525759	−0.262879	−	0.964829i	$-0.584672\pi$
$907$	−15.8340	−0.525759	−0.262879	+	0.964829i	$0.584672\pi$
$908$	2.41699	0.0802108
$909$	0	0
$910$	0	0
$911$	18.3948	0.609446	0.304723	−	0.952441i	$-0.401436\pi$
$911$	18.3948	0.609446	0.304723	+	0.952441i	$0.401436\pi$
$912$	0	0
$913$	−15.5830	−0.515722
$914$	−40.5830	−1.34237
$915$	0	0
$916$	−6.70850	−0.221655
$917$	0	0
$918$	0	0
$919$	9.35425	0.308568	0.154284	−	0.988027i	$-0.450693\pi$
$919$	9.35425	0.308568	0.154284	+	0.988027i	$0.450693\pi$
$920$	7.00000	0.230783
$921$	0	0
$922$	−24.5830	−0.809598
$923$	10.8340	0.356605
$924$	0	0
$925$	2.58301	0.0849287
$926$	39.0405	1.28295
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−34.4575	−1.13051	−0.565257	−	0.824915i	$-0.691223\pi$
$929$	−34.4575	−1.13051	−0.565257	+	0.824915i	$0.691223\pi$
$930$	0	0
$931$	0	0
$932$	−20.1660	−0.660560
$933$	0	0
$934$	8.00000	0.261768
$935$	7.93725	0.259576
$936$	0	0
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	0	0
$939$	0	0
$940$	−10.4170	−0.339765
$941$	3.41699	0.111391	0.0556954	−	0.998448i	$-0.482262\pi$
$941$	3.41699	0.111391	0.0556954	+	0.998448i	$0.482262\pi$
$942$	0	0
$943$	23.8118	0.775418
$944$	6.58301	0.214259
$945$	0	0
$946$	−9.29150	−0.302093
$947$	15.4170	0.500985	0.250493	−	0.968119i	$-0.419407\pi$
$947$	15.4170	0.500985	0.250493	+	0.968119i	$0.419407\pi$
$948$	0	0
$949$	61.1660	1.98553
$950$	−10.5830	−0.343358
$951$	0	0
$952$	0	0
$953$	−26.7490	−0.866486	−0.433243	−	0.901277i	$-0.642631\pi$
$953$	−26.7490	−0.866486	−0.433243	+	0.901277i	$0.642631\pi$
$954$	0	0
$955$	−17.7490	−0.574345
$956$	−2.70850	−0.0875991
$957$	0	0
$958$	−33.8745	−1.09444
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	5.16601	0.166559
$963$	0	0
$964$	27.1660	0.874958
$965$	−42.0000	−1.35203
$966$	0	0
$967$	40.0627	1.28833	0.644166	−	0.764886i	$-0.277205\pi$
$967$	40.0627	1.28833	0.644166	+	0.764886i	$0.277205\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	30.6458	0.983976
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	0	0
$974$	21.8745	0.700904
$975$	0	0
$976$	−11.9373	−0.382102
$977$	−13.1660	−0.421218	−0.210609	−	0.977570i	$-0.567545\pi$
$977$	−13.1660	−0.421218	−0.210609	+	0.977570i	$0.567545\pi$
$978$	0	0
$979$	−2.70850	−0.0865640
$980$	0	0
$981$	0	0
$982$	1.00000	0.0319113
$983$	42.5203	1.35619	0.678093	−	0.734976i	$-0.262807\pi$
$983$	42.5203	1.35619	0.678093	+	0.734976i	$0.262807\pi$
$984$	0	0
$985$	−57.8745	−1.84404
$986$	6.00000	0.191079
$987$	0	0
$988$	−21.1660	−0.673380
$989$	24.5830	0.781694
$990$	0	0
$991$	−43.8745	−1.39372	−0.696860	−	0.717207i	$-0.745420\pi$
$991$	−43.8745	−1.39372	−0.696860	+	0.717207i	$0.745420\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	0	0
$995$	50.7085	1.60757
$996$	0	0
$997$	−6.83399	−0.216435	−0.108217	−	0.994127i	$-0.534514\pi$
$997$	−6.83399	−0.216435	−0.108217	+	0.994127i	$0.534514\pi$
$998$	−18.5830	−0.588235
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dm.1.2		2
3.2	odd	2		3234.2.a.ba.1.1		2
7.3	odd	6		1386.2.k.r.793.2		4
7.5	odd	6		1386.2.k.r.991.2		4
7.6	odd	2		9702.2.a.db.1.1		2
21.5	even	6		462.2.i.e.67.1	✓	4
21.17	even	6		462.2.i.e.331.1	yes	4
21.20	even	2		3234.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.e.67.1	✓	4	21.5	even	6
462.2.i.e.331.1	yes	4	21.17	even	6
1386.2.k.r.793.2		4	7.3	odd	6
1386.2.k.r.991.2		4	7.5	odd	6
3234.2.a.w.1.2		2	21.20	even	2
3234.2.a.ba.1.1		2	3.2	odd	2
9702.2.a.db.1.1		2	7.6	odd	2
9702.2.a.dm.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.64575 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.64575 q^{5} +1.00000 q^{8} +2.64575 q^{10} -1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -5.29150 q^{19} +2.64575 q^{20} -1.00000 q^{22} +2.64575 q^{23} +2.00000 q^{25} +4.00000 q^{26} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +1.29150 q^{37} -5.29150 q^{38} +2.64575 q^{40} +9.00000 q^{41} +9.29150 q^{43} -1.00000 q^{44} +2.64575 q^{46} -3.93725 q^{47} +2.00000 q^{50} +4.00000 q^{52} -4.00000 q^{53} -2.64575 q^{55} -2.00000 q^{58} +6.58301 q^{59} -11.9373 q^{61} +4.00000 q^{62} +1.00000 q^{64} +10.5830 q^{65} +7.58301 q^{67} -3.00000 q^{68} +2.70850 q^{71} +15.2915 q^{73} +1.29150 q^{74} -5.29150 q^{76} +10.6458 q^{79} +2.64575 q^{80} +9.00000 q^{82} +15.5830 q^{83} -7.93725 q^{85} +9.29150 q^{86} -1.00000 q^{88} +2.70850 q^{89} +2.64575 q^{92} -3.93725 q^{94} -14.0000 q^{95} +11.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{22} + 4 q^{25} + 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{34} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 2 q^{44} + 8 q^{47} + 4 q^{50} + 8 q^{52} - 8 q^{53} - 4 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{67} - 6 q^{68} + 16 q^{71} + 20 q^{73} - 8 q^{74} + 16 q^{79} + 18 q^{82} + 10 q^{83} + 8 q^{86} - 2 q^{88} + 16 q^{89} + 8 q^{94} - 28 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 8 * q^13 + 2 * q^16 - 6 * q^17 - 2 * q^22 + 4 * q^25 + 8 * q^26 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^34 - 8 * q^37 + 18 * q^41 + 8 * q^43 - 2 * q^44 + 8 * q^47 + 4 * q^50 + 8 * q^52 - 8 * q^53 - 4 * q^58 - 8 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 - 6 * q^67 - 6 * q^68 + 16 * q^71 + 20 * q^73 - 8 * q^74 + 16 * q^79 + 18 * q^82 + 10 * q^83 + 8 * q^86 - 2 * q^88 + 16 * q^89 + 8 * q^94 - 28 * q^95 + 2 * q^97

Introduction

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