LMFDB - Embedded newform 9702.2.a.dt.1.3

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:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 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			1.3
Root			$-0.523976$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{8} -3.20147 q^{10} -1.00000 q^{11} -6.40294 q^{13} +1.00000 q^{16} -1.00000 q^{17} +1.15352 q^{19} +3.20147 q^{20} +1.00000 q^{22} -1.95205 q^{23} +5.24943 q^{25} +6.40294 q^{26} -7.24943 q^{29} +10.4989 q^{31} -1.00000 q^{32} +1.00000 q^{34} +5.15352 q^{37} -1.15352 q^{38} -3.20147 q^{40} -8.24943 q^{41} +5.15352 q^{43} -1.00000 q^{44} +1.95205 q^{46} +6.04795 q^{47} -5.24943 q^{50} -6.40294 q^{52} +6.40294 q^{53} -3.20147 q^{55} +7.24943 q^{58} -11.5565 q^{59} +9.70032 q^{61} -10.4989 q^{62} +1.00000 q^{64} -20.4989 q^{65} +6.24943 q^{67} -1.00000 q^{68} -5.24943 q^{71} -2.09591 q^{73} -5.15352 q^{74} +1.15352 q^{76} -5.60442 q^{79} +3.20147 q^{80} +8.24943 q^{82} -6.55646 q^{83} -3.20147 q^{85} -5.15352 q^{86} +1.00000 q^{88} +18.4029 q^{89} -1.95205 q^{92} -6.04795 q^{94} +3.69296 q^{95} -5.49885 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 3 q^{11} + 3 q^{16} - 3 q^{17} - 3 q^{19} + 3 q^{22} - 9 q^{23} + 3 q^{25} - 9 q^{29} + 6 q^{31} - 3 q^{32} + 3 q^{34} + 9 q^{37} + 3 q^{38} - 12 q^{41} + 9 q^{43} - 3 q^{44} + 9 q^{46} + 15 q^{47} - 3 q^{50} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} - 36 q^{65} + 6 q^{67} - 3 q^{68} - 3 q^{71} - 9 q^{74} - 3 q^{76} + 12 q^{79} + 12 q^{82} + 6 q^{83} - 9 q^{86} + 3 q^{88} + 36 q^{89} - 9 q^{92} - 15 q^{94} + 24 q^{95} + 9 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 - 3 * q^11 + 3 * q^16 - 3 * q^17 - 3 * q^19 + 3 * q^22 - 9 * q^23 + 3 * q^25 - 9 * q^29 + 6 * q^31 - 3 * q^32 + 3 * q^34 + 9 * q^37 + 3 * q^38 - 12 * q^41 + 9 * q^43 - 3 * q^44 + 9 * q^46 + 15 * q^47 - 3 * q^50 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 - 36 * q^65 + 6 * q^67 - 3 * q^68 - 3 * q^71 - 9 * q^74 - 3 * q^76 + 12 * q^79 + 12 * q^82 + 6 * q^83 - 9 * q^86 + 3 * q^88 + 36 * q^89 - 9 * q^92 - 15 * q^94 + 24 * q^95 + 9 * 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
$5$	3.20147	1.43174	0.715871	−	0.698233i	$-0.246030\pi$
$5$	3.20147	1.43174	0.715871	+	0.698233i	$0.246030\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.20147	−1.01239
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.40294	−1.77586	−0.887929	−	0.459981i	$-0.847856\pi$
$13$	−6.40294	−1.77586	−0.887929	+	0.459981i	$0.847856\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	1.15352	0.264636	0.132318	−	0.991207i	$-0.457758\pi$
$19$	1.15352	0.264636	0.132318	+	0.991207i	$0.457758\pi$
$20$	3.20147	0.715871
$21$	0	0
$22$	1.00000	0.213201
$23$	−1.95205	−0.407030	−0.203515	−	0.979072i	$-0.565237\pi$
$23$	−1.95205	−0.407030	−0.203515	+	0.979072i	$0.565237\pi$
$24$	0	0
$25$	5.24943	1.04989
$26$	6.40294	1.25572
$27$	0	0
$28$	0	0
$29$	−7.24943	−1.34618	−0.673092	−	0.739559i	$-0.735034\pi$
$29$	−7.24943	−1.34618	−0.673092	+	0.739559i	$0.735034\pi$
$30$	0	0
$31$	10.4989	1.88565	0.942825	−	0.333289i	$-0.108159\pi$
$31$	10.4989	1.88565	0.942825	+	0.333289i	$0.108159\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	5.15352	0.847233	0.423617	−	0.905842i	$-0.360760\pi$
$37$	5.15352	0.847233	0.423617	+	0.905842i	$0.360760\pi$
$38$	−1.15352	−0.187126
$39$	0	0
$40$	−3.20147	−0.506197
$41$	−8.24943	−1.28834	−0.644172	−	0.764881i	$-0.722798\pi$
$41$	−8.24943	−1.28834	−0.644172	+	0.764881i	$0.722798\pi$
$42$	0	0
$43$	5.15352	0.785904	0.392952	−	0.919559i	$-0.371454\pi$
$43$	5.15352	0.785904	0.392952	+	0.919559i	$0.371454\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	1.95205	0.287814
$47$	6.04795	0.882185	0.441092	−	0.897462i	$-0.354591\pi$
$47$	6.04795	0.882185	0.441092	+	0.897462i	$0.354591\pi$
$48$	0	0
$49$	0	0
$50$	−5.24943	−0.742381
$51$	0	0
$52$	−6.40294	−0.887929
$53$	6.40294	0.879512	0.439756	−	0.898117i	$-0.355065\pi$
$53$	6.40294	0.879512	0.439756	+	0.898117i	$0.355065\pi$
$54$	0	0
$55$	−3.20147	−0.431686
$56$	0	0
$57$	0	0
$58$	7.24943	0.951896
$59$	−11.5565	−1.50452	−0.752262	−	0.658864i	$-0.771037\pi$
$59$	−11.5565	−1.50452	−0.752262	+	0.658864i	$0.771037\pi$
$60$	0	0
$61$	9.70032	1.24200	0.621000	−	0.783811i	$-0.286727\pi$
$61$	9.70032	1.24200	0.621000	+	0.783811i	$0.286727\pi$
$62$	−10.4989	−1.33336
$63$	0	0
$64$	1.00000	0.125000
$65$	−20.4989	−2.54257
$66$	0	0
$67$	6.24943	0.763489	0.381744	−	0.924268i	$-0.375323\pi$
$67$	6.24943	0.763489	0.381744	+	0.924268i	$0.375323\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	−5.24943	−0.622992	−0.311496	−	0.950247i	$-0.600830\pi$
$71$	−5.24943	−0.622992	−0.311496	+	0.950247i	$0.600830\pi$
$72$	0	0
$73$	−2.09591	−0.245307	−0.122654	−	0.992450i	$-0.539140\pi$
$73$	−2.09591	−0.245307	−0.122654	+	0.992450i	$0.539140\pi$
$74$	−5.15352	−0.599084
$75$	0	0
$76$	1.15352	0.132318
$77$	0	0
$78$	0	0
$79$	−5.60442	−0.630546	−0.315273	−	0.949001i	$-0.602096\pi$
$79$	−5.60442	−0.630546	−0.315273	+	0.949001i	$0.602096\pi$
$80$	3.20147	0.357935
$81$	0	0
$82$	8.24943	0.910997
$83$	−6.55646	−0.719665	−0.359833	−	0.933017i	$-0.617166\pi$
$83$	−6.55646	−0.719665	−0.359833	+	0.933017i	$0.617166\pi$
$84$	0	0
$85$	−3.20147	−0.347248
$86$	−5.15352	−0.555718
$87$	0	0
$88$	1.00000	0.106600
$89$	18.4029	1.95071	0.975354	−	0.220645i	$-0.0708163\pi$
$89$	18.4029	1.95071	0.975354	+	0.220645i	$0.0708163\pi$
$90$	0	0
$91$	0	0
$92$	−1.95205	−0.203515
$93$	0	0
$94$	−6.04795	−0.623799
$95$	3.69296	0.378890
$96$	0	0
$97$	−5.49885	−0.558324	−0.279162	−	0.960244i	$-0.590057\pi$
$97$	−5.49885	−0.558324	−0.279162	+	0.960244i	$0.590057\pi$
$98$	0	0
$99$	0	0
$100$	5.24943	0.524943
$101$	19.9594	1.98604	0.993018	−	0.117965i	$-0.0376372\pi$
$101$	19.9594	1.98604	0.993018	+	0.117965i	$0.0376372\pi$
$102$	0	0
$103$	0.307039	0.0302535	0.0151267	−	0.999886i	$-0.495185\pi$
$103$	0.307039	0.0302535	0.0151267	+	0.999886i	$0.495185\pi$
$104$	6.40294	0.627860
$105$	0	0
$106$	−6.40294	−0.621909
$107$	9.05531	0.875410	0.437705	−	0.899119i	$-0.355791\pi$
$107$	9.05531	0.875410	0.437705	+	0.899119i	$0.355791\pi$
$108$	0	0
$109$	−9.70032	−0.929122	−0.464561	−	0.885541i	$-0.653788\pi$
$109$	−9.70032	−0.929122	−0.464561	+	0.885541i	$0.653788\pi$
$110$	3.20147	0.305248
$111$	0	0
$112$	0	0
$113$	12.4989	1.17579	0.587896	−	0.808936i	$-0.299956\pi$
$113$	12.4989	1.17579	0.587896	+	0.808936i	$0.299956\pi$
$114$	0	0
$115$	−6.24943	−0.582762
$116$	−7.24943	−0.673092
$117$	0	0
$118$	11.5565	1.06386
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.70032	−0.878226
$123$	0	0
$124$	10.4989	0.942825
$125$	0.798528	0.0714225
$126$	0	0
$127$	8.35499	0.741386	0.370693	−	0.928756i	$-0.379120\pi$
$127$	8.35499	0.741386	0.370693	+	0.928756i	$0.379120\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	20.4989	1.79787
$131$	−15.3047	−1.33718	−0.668591	−	0.743631i	$-0.733102\pi$
$131$	−15.3047	−1.33718	−0.668591	+	0.743631i	$0.733102\pi$
$132$	0	0
$133$	0	0
$134$	−6.24943	−0.539868
$135$	0	0
$136$	1.00000	0.0857493
$137$	6.40294	0.547040	0.273520	−	0.961866i	$-0.411812\pi$
$137$	6.40294	0.547040	0.273520	+	0.961866i	$0.411812\pi$
$138$	0	0
$139$	21.7483	1.84466	0.922332	−	0.386398i	$-0.126281\pi$
$139$	21.7483	1.84466	0.922332	+	0.386398i	$0.126281\pi$
$140$	0	0
$141$	0	0
$142$	5.24943	0.440522
$143$	6.40294	0.535441
$144$	0	0
$145$	−23.2088	−1.92739
$146$	2.09591	0.173458
$147$	0	0
$148$	5.15352	0.423617
$149$	−6.84648	−0.560886	−0.280443	−	0.959871i	$-0.590481\pi$
$149$	−6.84648	−0.560886	−0.280443	+	0.959871i	$0.590481\pi$
$150$	0	0
$151$	14.8538	1.20879	0.604394	−	0.796685i	$-0.293415\pi$
$151$	14.8538	1.20879	0.604394	+	0.796685i	$0.293415\pi$
$152$	−1.15352	−0.0935628
$153$	0	0
$154$	0	0
$155$	33.6118	2.69976
$156$	0	0
$157$	−9.46056	−0.755035	−0.377517	−	0.926002i	$-0.623222\pi$
$157$	−9.46056	−0.755035	−0.377517	+	0.926002i	$0.623222\pi$
$158$	5.60442	0.445863
$159$	0	0
$160$	−3.20147	−0.253099
$161$	0	0
$162$	0	0
$163$	13.0553	1.02257	0.511286	−	0.859411i	$-0.329169\pi$
$163$	13.0553	1.02257	0.511286	+	0.859411i	$0.329169\pi$
$164$	−8.24943	−0.644172
$165$	0	0
$166$	6.55646	0.508880
$167$	6.70998	0.519234	0.259617	−	0.965712i	$-0.416404\pi$
$167$	6.70998	0.519234	0.259617	+	0.965712i	$0.416404\pi$
$168$	0	0
$169$	27.9977	2.15367
$170$	3.20147	0.245542
$171$	0	0
$172$	5.15352	0.392952
$173$	5.50115	0.418245	0.209122	−	0.977889i	$-0.432939\pi$
$173$	5.50115	0.418245	0.209122	+	0.977889i	$0.432939\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−18.4029	−1.37936
$179$	−1.65237	−0.123504	−0.0617520	−	0.998092i	$-0.519669\pi$
$179$	−1.65237	−0.123504	−0.0617520	+	0.998092i	$0.519669\pi$
$180$	0	0
$181$	24.9018	1.85094	0.925468	−	0.378826i	$-0.123672\pi$
$181$	24.9018	1.85094	0.925468	+	0.378826i	$0.123672\pi$
$182$	0	0
$183$	0	0
$184$	1.95205	0.143907
$185$	16.4989	1.21302
$186$	0	0
$187$	1.00000	0.0731272
$188$	6.04795	0.441092
$189$	0	0
$190$	−3.69296	−0.267916
$191$	4.80589	0.347742	0.173871	−	0.984768i	$-0.444372\pi$
$191$	4.80589	0.347742	0.173871	+	0.984768i	$0.444372\pi$
$192$	0	0
$193$	8.70998	0.626958	0.313479	−	0.949595i	$-0.398505\pi$
$193$	8.70998	0.626958	0.313479	+	0.949595i	$0.398505\pi$
$194$	5.49885	0.394794
$195$	0	0
$196$	0	0
$197$	−0.654669	−0.0466433	−0.0233216	−	0.999728i	$-0.507424\pi$
$197$	−0.654669	−0.0466433	−0.0233216	+	0.999728i	$0.507424\pi$
$198$	0	0
$199$	−7.69296	−0.545340	−0.272670	−	0.962108i	$-0.587907\pi$
$199$	−7.69296	−0.545340	−0.272670	+	0.962108i	$0.587907\pi$
$200$	−5.24943	−0.371190
$201$	0	0
$202$	−19.9594	−1.40434
$203$	0	0
$204$	0	0
$205$	−26.4103	−1.84458
$206$	−0.307039	−0.0213924
$207$	0	0
$208$	−6.40294	−0.443964
$209$	−1.15352	−0.0797906
$210$	0	0
$211$	14.5948	1.00474	0.502372	−	0.864651i	$-0.332461\pi$
$211$	14.5948	1.00474	0.502372	+	0.864651i	$0.332461\pi$
$212$	6.40294	0.439756
$213$	0	0
$214$	−9.05531	−0.619009
$215$	16.4989	1.12521
$216$	0	0
$217$	0	0
$218$	9.70032	0.656989
$219$	0	0
$220$	−3.20147	−0.215843
$221$	6.40294	0.430709
$222$	0	0
$223$	−14.9018	−0.997898	−0.498949	−	0.866631i	$-0.666281\pi$
$223$	−14.9018	−0.997898	−0.498949	+	0.866631i	$0.666281\pi$
$224$	0	0
$225$	0	0
$226$	−12.4989	−0.831411
$227$	0.249425	0.0165549	0.00827746	−	0.999966i	$-0.497365\pi$
$227$	0.249425	0.0165549	0.00827746	+	0.999966i	$0.497365\pi$
$228$	0	0
$229$	−4.09591	−0.270665	−0.135333	−	0.990800i	$-0.543210\pi$
$229$	−4.09591	−0.270665	−0.135333	+	0.990800i	$0.543210\pi$
$230$	6.24943	0.412075
$231$	0	0
$232$	7.24943	0.475948
$233$	−11.3070	−0.740749	−0.370374	−	0.928883i	$-0.620771\pi$
$233$	−11.3070	−0.740749	−0.370374	+	0.928883i	$0.620771\pi$
$234$	0	0
$235$	19.3624	1.26306
$236$	−11.5565	−0.752262
$237$	0	0
$238$	0	0
$239$	−6.59476	−0.426579	−0.213290	−	0.976989i	$-0.568418\pi$
$239$	−6.59476	−0.426579	−0.213290	+	0.976989i	$0.568418\pi$
$240$	0	0
$241$	17.1129	1.10234	0.551170	−	0.834393i	$-0.314181\pi$
$241$	17.1129	1.10234	0.551170	+	0.834393i	$0.314181\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	9.70032	0.621000
$245$	0	0
$246$	0	0
$247$	−7.38592	−0.469955
$248$	−10.4989	−0.666678
$249$	0	0
$250$	−0.798528	−0.0505033
$251$	−7.95941	−0.502393	−0.251197	−	0.967936i	$-0.580824\pi$
$251$	−7.95941	−0.502393	−0.251197	+	0.967936i	$0.580824\pi$
$252$	0	0
$253$	1.95205	0.122724
$254$	−8.35499	−0.524239
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.4029	1.39746	0.698729	−	0.715387i	$-0.253749\pi$
$257$	22.4029	1.39746	0.698729	+	0.715387i	$0.253749\pi$
$258$	0	0
$259$	0	0
$260$	−20.4989	−1.27128
$261$	0	0
$262$	15.3047	0.945530
$263$	25.0170	1.54262	0.771308	−	0.636462i	$-0.219603\pi$
$263$	25.0170	1.54262	0.771308	+	0.636462i	$0.219603\pi$
$264$	0	0
$265$	20.4989	1.25923
$266$	0	0
$267$	0	0
$268$	6.24943	0.381744
$269$	−8.10327	−0.494065	−0.247032	−	0.969007i	$-0.579455\pi$
$269$	−8.10327	−0.494065	−0.247032	+	0.969007i	$0.579455\pi$
$270$	0	0
$271$	−4.80589	−0.291937	−0.145968	−	0.989289i	$-0.546630\pi$
$271$	−4.80589	−0.291937	−0.145968	+	0.989289i	$0.546630\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−6.40294	−0.386816
$275$	−5.24943	−0.316552
$276$	0	0
$277$	−11.2088	−0.673474	−0.336737	−	0.941599i	$-0.609323\pi$
$277$	−11.2088	−0.673474	−0.336737	+	0.941599i	$0.609323\pi$
$278$	−21.7483	−1.30437
$279$	0	0
$280$	0	0
$281$	17.4989	1.04389	0.521947	−	0.852978i	$-0.325206\pi$
$281$	17.4989	1.04389	0.521947	+	0.852978i	$0.325206\pi$
$282$	0	0
$283$	27.4006	1.62880	0.814400	−	0.580304i	$-0.197066\pi$
$283$	27.4006	1.62880	0.814400	+	0.580304i	$0.197066\pi$
$284$	−5.24943	−0.311496
$285$	0	0
$286$	−6.40294	−0.378614
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	23.2088	1.36287
$291$	0	0
$292$	−2.09591	−0.122654
$293$	19.7483	1.15371	0.576853	−	0.816848i	$-0.304280\pi$
$293$	19.7483	1.15371	0.576853	+	0.816848i	$0.304280\pi$
$294$	0	0
$295$	−36.9977	−2.15409
$296$	−5.15352	−0.299542
$297$	0	0
$298$	6.84648	0.396606
$299$	12.4989	0.722827
$300$	0	0
$301$	0	0
$302$	−14.8538	−0.854743
$303$	0	0
$304$	1.15352	0.0661589
$305$	31.0553	1.77822
$306$	0	0
$307$	4.30704	0.245816	0.122908	−	0.992418i	$-0.460778\pi$
$307$	4.30704	0.245816	0.122908	+	0.992418i	$0.460778\pi$
$308$	0	0
$309$	0	0
$310$	−33.6118	−1.90902
$311$	14.7579	0.836846	0.418423	−	0.908252i	$-0.362583\pi$
$311$	14.7579	0.836846	0.418423	+	0.908252i	$0.362583\pi$
$312$	0	0
$313$	20.7506	1.17289	0.586446	−	0.809988i	$-0.300527\pi$
$313$	20.7506	1.17289	0.586446	+	0.809988i	$0.300527\pi$
$314$	9.46056	0.533890
$315$	0	0
$316$	−5.60442	−0.315273
$317$	−16.1992	−0.909836	−0.454918	−	0.890533i	$-0.650331\pi$
$317$	−16.1992	−0.909836	−0.454918	+	0.890533i	$0.650331\pi$
$318$	0	0
$319$	7.24943	0.405890
$320$	3.20147	0.178968
$321$	0	0
$322$	0	0
$323$	−1.15352	−0.0641835
$324$	0	0
$325$	−33.6118	−1.86445
$326$	−13.0553	−0.723067
$327$	0	0
$328$	8.24943	0.455498
$329$	0	0
$330$	0	0
$331$	21.2471	1.16785	0.583924	−	0.811808i	$-0.301517\pi$
$331$	21.2471	1.16785	0.583924	+	0.811808i	$0.301517\pi$
$332$	−6.55646	−0.359833
$333$	0	0
$334$	−6.70998	−0.367154
$335$	20.0074	1.09312
$336$	0	0
$337$	−25.5159	−1.38994	−0.694969	−	0.719040i	$-0.744582\pi$
$337$	−25.5159	−1.38994	−0.694969	+	0.719040i	$0.744582\pi$
$338$	−27.9977	−1.52287
$339$	0	0
$340$	−3.20147	−0.173624
$341$	−10.4989	−0.568545
$342$	0	0
$343$	0	0
$344$	−5.15352	−0.277859
$345$	0	0
$346$	−5.50115	−0.295744
$347$	−3.05531	−0.164018	−0.0820089	−	0.996632i	$-0.526134\pi$
$347$	−3.05531	−0.164018	−0.0820089	+	0.996632i	$0.526134\pi$
$348$	0	0
$349$	−4.00736	−0.214509	−0.107255	−	0.994232i	$-0.534206\pi$
$349$	−4.00736	−0.214509	−0.107255	+	0.994232i	$0.534206\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	1.09821	0.0584516	0.0292258	−	0.999573i	$-0.490696\pi$
$353$	1.09821	0.0584516	0.0292258	+	0.999573i	$0.490696\pi$
$354$	0	0
$355$	−16.8059	−0.891964
$356$	18.4029	0.975354
$357$	0	0
$358$	1.65237	0.0873305
$359$	−15.6118	−0.823958	−0.411979	−	0.911193i	$-0.635162\pi$
$359$	−15.6118	−0.823958	−0.411979	+	0.911193i	$0.635162\pi$
$360$	0	0
$361$	−17.6694	−0.929968
$362$	−24.9018	−1.30881
$363$	0	0
$364$	0	0
$365$	−6.70998	−0.351217
$366$	0	0
$367$	0.791166	0.0412985	0.0206493	−	0.999787i	$-0.493427\pi$
$367$	0.791166	0.0412985	0.0206493	+	0.999787i	$0.493427\pi$
$368$	−1.95205	−0.101757
$369$	0	0
$370$	−16.4989	−0.857734
$371$	0	0
$372$	0	0
$373$	15.9115	0.823864	0.411932	−	0.911215i	$-0.364854\pi$
$373$	15.9115	0.823864	0.411932	+	0.911215i	$0.364854\pi$
$374$	−1.00000	−0.0517088
$375$	0	0
$376$	−6.04795	−0.311899
$377$	46.4177	2.39063
$378$	0	0
$379$	−17.0553	−0.876073	−0.438036	−	0.898957i	$-0.644326\pi$
$379$	−17.0553	−0.876073	−0.438036	+	0.898957i	$0.644326\pi$
$380$	3.69296	0.189445
$381$	0	0
$382$	−4.80589	−0.245891
$383$	−13.4412	−0.686815	−0.343408	−	0.939186i	$-0.611581\pi$
$383$	−13.4412	−0.686815	−0.343408	+	0.939186i	$0.611581\pi$
$384$	0	0
$385$	0	0
$386$	−8.70998	−0.443327
$387$	0	0
$388$	−5.49885	−0.279162
$389$	7.20147	0.365129	0.182565	−	0.983194i	$-0.441560\pi$
$389$	7.20147	0.365129	0.182565	+	0.983194i	$0.441560\pi$
$390$	0	0
$391$	1.95205	0.0987193
$392$	0	0
$393$	0	0
$394$	0.654669	0.0329818
$395$	−17.9424	−0.902779
$396$	0	0
$397$	12.9424	0.649560	0.324780	−	0.945790i	$-0.394710\pi$
$397$	12.9424	0.649560	0.324780	+	0.945790i	$0.394710\pi$
$398$	7.69296	0.385613
$399$	0	0
$400$	5.24943	0.262471
$401$	16.9018	0.844035	0.422018	−	0.906588i	$-0.361322\pi$
$401$	16.9018	0.844035	0.422018	+	0.906588i	$0.361322\pi$
$402$	0	0
$403$	−67.2236	−3.34864
$404$	19.9594	0.993018
$405$	0	0
$406$	0	0
$407$	−5.15352	−0.255450
$408$	0	0
$409$	−37.0936	−1.83416	−0.917080	−	0.398702i	$-0.869461\pi$
$409$	−37.0936	−1.83416	−0.917080	+	0.398702i	$0.869461\pi$
$410$	26.4103	1.30431
$411$	0	0
$412$	0.307039	0.0151267
$413$	0	0
$414$	0	0
$415$	−20.9903	−1.03038
$416$	6.40294	0.313930
$417$	0	0
$418$	1.15352	0.0564205
$419$	−22.4583	−1.09716	−0.548579	−	0.836099i	$-0.684831\pi$
$419$	−22.4583	−1.09716	−0.548579	+	0.836099i	$0.684831\pi$
$420$	0	0
$421$	−23.3453	−1.13778	−0.568891	−	0.822413i	$-0.692627\pi$
$421$	−23.3453	−1.13778	−0.568891	+	0.822413i	$0.692627\pi$
$422$	−14.5948	−0.710462
$423$	0	0
$424$	−6.40294	−0.310954
$425$	−5.24943	−0.254635
$426$	0	0
$427$	0	0
$428$	9.05531	0.437705
$429$	0	0
$430$	−16.4989	−0.795645
$431$	−31.8229	−1.53286	−0.766428	−	0.642330i	$-0.777968\pi$
$431$	−31.8229	−1.53286	−0.766428	+	0.642330i	$0.777968\pi$
$432$	0	0
$433$	30.3047	1.45635	0.728176	−	0.685390i	$-0.240368\pi$
$433$	30.3047	1.45635	0.728176	+	0.685390i	$0.240368\pi$
$434$	0	0
$435$	0	0
$436$	−9.70032	−0.464561
$437$	−2.25172	−0.107715
$438$	0	0
$439$	−10.6620	−0.508871	−0.254435	−	0.967090i	$-0.581890\pi$
$439$	−10.6620	−0.508871	−0.254435	+	0.967090i	$0.581890\pi$
$440$	3.20147	0.152624
$441$	0	0
$442$	−6.40294	−0.304557
$443$	36.0553	1.71304	0.856520	−	0.516114i	$-0.172622\pi$
$443$	36.0553	1.71304	0.856520	+	0.516114i	$0.172622\pi$
$444$	0	0
$445$	58.9165	2.79291
$446$	14.9018	0.705620
$447$	0	0
$448$	0	0
$449$	−23.4006	−1.10434	−0.552172	−	0.833730i	$-0.686201\pi$
$449$	−23.4006	−1.10434	−0.552172	+	0.833730i	$0.686201\pi$
$450$	0	0
$451$	8.24943	0.388450
$452$	12.4989	0.587896
$453$	0	0
$454$	−0.249425	−0.0117061
$455$	0	0
$456$	0	0
$457$	−18.6141	−0.870730	−0.435365	−	0.900254i	$-0.643381\pi$
$457$	−18.6141	−0.870730	−0.435365	+	0.900254i	$0.643381\pi$
$458$	4.09591	0.191389
$459$	0	0
$460$	−6.24943	−0.291381
$461$	31.4412	1.46436	0.732182	−	0.681109i	$-0.238502\pi$
$461$	31.4412	1.46436	0.732182	+	0.681109i	$0.238502\pi$
$462$	0	0
$463$	−22.0959	−1.02688	−0.513442	−	0.858124i	$-0.671630\pi$
$463$	−22.0959	−1.02688	−0.513442	+	0.858124i	$0.671630\pi$
$464$	−7.24943	−0.336546
$465$	0	0
$466$	11.3070	0.523788
$467$	−21.0576	−0.974430	−0.487215	−	0.873282i	$-0.661987\pi$
$467$	−21.0576	−0.974430	−0.487215	+	0.873282i	$0.661987\pi$
$468$	0	0
$469$	0	0
$470$	−19.3624	−0.893119
$471$	0	0
$472$	11.5565	0.531929
$473$	−5.15352	−0.236959
$474$	0	0
$475$	6.05531	0.277837
$476$	0	0
$477$	0	0
$478$	6.59476	0.301637
$479$	21.2088	0.969056	0.484528	−	0.874776i	$-0.338991\pi$
$479$	21.2088	0.969056	0.484528	+	0.874776i	$0.338991\pi$
$480$	0	0
$481$	−32.9977	−1.50457
$482$	−17.1129	−0.779473
$483$	0	0
$484$	1.00000	0.0454545
$485$	−17.6044	−0.799375
$486$	0	0
$487$	−38.7100	−1.75412	−0.877058	−	0.480384i	$-0.840497\pi$
$487$	−38.7100	−1.75412	−0.877058	+	0.480384i	$0.840497\pi$
$488$	−9.70032	−0.439113
$489$	0	0
$490$	0	0
$491$	−19.1705	−0.865154	−0.432577	−	0.901597i	$-0.642396\pi$
$491$	−19.1705	−0.865154	−0.432577	+	0.901597i	$0.642396\pi$
$492$	0	0
$493$	7.24943	0.326498
$494$	7.38592	0.332308
$495$	0	0
$496$	10.4989	0.471412
$497$	0	0
$498$	0	0
$499$	−27.9188	−1.24982	−0.624909	−	0.780698i	$-0.714864\pi$
$499$	−27.9188	−1.24982	−0.624909	+	0.780698i	$0.714864\pi$
$500$	0.798528	0.0357112
$501$	0	0
$502$	7.95941	0.355246
$503$	8.70998	0.388359	0.194179	−	0.980966i	$-0.437796\pi$
$503$	8.70998	0.388359	0.194179	+	0.980966i	$0.437796\pi$
$504$	0	0
$505$	63.8995	2.84349
$506$	−1.95205	−0.0867791
$507$	0	0
$508$	8.35499	0.370693
$509$	1.78887	0.0792901	0.0396451	−	0.999214i	$-0.487377\pi$
$509$	1.78887	0.0792901	0.0396451	+	0.999214i	$0.487377\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.4029	−0.988152
$515$	0.982977	0.0433151
$516$	0	0
$517$	−6.04795	−0.265989
$518$	0	0
$519$	0	0
$520$	20.4989	0.898934
$521$	−23.6118	−1.03445	−0.517225	−	0.855849i	$-0.673035\pi$
$521$	−23.6118	−1.03445	−0.517225	+	0.855849i	$0.673035\pi$
$522$	0	0
$523$	43.4006	1.89778	0.948889	−	0.315610i	$-0.102209\pi$
$523$	43.4006	1.89778	0.948889	+	0.315610i	$0.102209\pi$
$524$	−15.3047	−0.668591
$525$	0	0
$526$	−25.0170	−1.09079
$527$	−10.4989	−0.457337
$528$	0	0
$529$	−19.1895	−0.834327
$530$	−20.4989	−0.890413
$531$	0	0
$532$	0	0
$533$	52.8206	2.28791
$534$	0	0
$535$	28.9903	1.25336
$536$	−6.24943	−0.269934
$537$	0	0
$538$	8.10327	0.349357
$539$	0	0
$540$	0	0
$541$	16.9903	0.730472	0.365236	−	0.930915i	$-0.380988\pi$
$541$	16.9903	0.730472	0.365236	+	0.930915i	$0.380988\pi$
$542$	4.80589	0.206431
$543$	0	0
$544$	1.00000	0.0428746
$545$	−31.0553	−1.33026
$546$	0	0
$547$	−1.34533	−0.0575222	−0.0287611	−	0.999586i	$-0.509156\pi$
$547$	−1.34533	−0.0575222	−0.0287611	+	0.999586i	$0.509156\pi$
$548$	6.40294	0.273520
$549$	0	0
$550$	5.24943	0.223836
$551$	−8.36235	−0.356248
$552$	0	0
$553$	0	0
$554$	11.2088	0.476218
$555$	0	0
$556$	21.7483	0.922332
$557$	16.6547	0.705681	0.352840	−	0.935683i	$-0.385216\pi$
$557$	16.6547	0.705681	0.352840	+	0.935683i	$0.385216\pi$
$558$	0	0
$559$	−32.9977	−1.39565
$560$	0	0
$561$	0	0
$562$	−17.4989	−0.738144
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	40.0147	1.68343
$566$	−27.4006	−1.15174
$567$	0	0
$568$	5.24943	0.220261
$569$	10.5542	0.442454	0.221227	−	0.975222i	$-0.428994\pi$
$569$	10.5542	0.442454	0.221227	+	0.975222i	$0.428994\pi$
$570$	0	0
$571$	4.75057	0.198805	0.0994027	−	0.995047i	$-0.468307\pi$
$571$	4.75057	0.198805	0.0994027	+	0.995047i	$0.468307\pi$
$572$	6.40294	0.267721
$573$	0	0
$574$	0	0
$575$	−10.2471	−0.427335
$576$	0	0
$577$	37.5542	1.56340	0.781700	−	0.623654i	$-0.214353\pi$
$577$	37.5542	1.56340	0.781700	+	0.623654i	$0.214353\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	−23.2088	−0.963694
$581$	0	0
$582$	0	0
$583$	−6.40294	−0.265183
$584$	2.09591	0.0867292
$585$	0	0
$586$	−19.7483	−0.815794
$587$	17.0982	0.705718	0.352859	−	0.935676i	$-0.385209\pi$
$587$	17.0982	0.705718	0.352859	+	0.935676i	$0.385209\pi$
$588$	0	0
$589$	12.1106	0.499010
$590$	36.9977	1.52317
$591$	0	0
$592$	5.15352	0.211808
$593$	32.2471	1.32423	0.662115	−	0.749402i	$-0.269659\pi$
$593$	32.2471	1.32423	0.662115	+	0.749402i	$0.269659\pi$
$594$	0	0
$595$	0	0
$596$	−6.84648	−0.280443
$597$	0	0
$598$	−12.4989	−0.511116
$599$	−3.39328	−0.138646	−0.0693229	−	0.997594i	$-0.522084\pi$
$599$	−3.39328	−0.138646	−0.0693229	+	0.997594i	$0.522084\pi$
$600$	0	0
$601$	30.4989	1.24407	0.622037	−	0.782988i	$-0.286305\pi$
$601$	30.4989	1.24407	0.622037	+	0.782988i	$0.286305\pi$
$602$	0	0
$603$	0	0
$604$	14.8538	0.604394
$605$	3.20147	0.130158
$606$	0	0
$607$	12.9903	0.527262	0.263631	−	0.964624i	$-0.415080\pi$
$607$	12.9903	0.527262	0.263631	+	0.964624i	$0.415080\pi$
$608$	−1.15352	−0.0467814
$609$	0	0
$610$	−31.0553	−1.25739
$611$	−38.7247	−1.56663
$612$	0	0
$613$	20.1992	0.815837	0.407918	−	0.913018i	$-0.366255\pi$
$613$	20.1992	0.815837	0.407918	+	0.913018i	$0.366255\pi$
$614$	−4.30704	−0.173818
$615$	0	0
$616$	0	0
$617$	−17.0936	−0.688163	−0.344081	−	0.938940i	$-0.611810\pi$
$617$	−17.0936	−0.688163	−0.344081	+	0.938940i	$0.611810\pi$
$618$	0	0
$619$	41.6694	1.67483	0.837417	−	0.546564i	$-0.184065\pi$
$619$	41.6694	1.67483	0.837417	+	0.546564i	$0.184065\pi$
$620$	33.6118	1.34988
$621$	0	0
$622$	−14.7579	−0.591739
$623$	0	0
$624$	0	0
$625$	−23.6907	−0.947626
$626$	−20.7506	−0.829360
$627$	0	0
$628$	−9.46056	−0.377517
$629$	−5.15352	−0.205484
$630$	0	0
$631$	−23.8995	−0.951424	−0.475712	−	0.879601i	$-0.657810\pi$
$631$	−23.8995	−0.951424	−0.475712	+	0.879601i	$0.657810\pi$
$632$	5.60442	0.222932
$633$	0	0
$634$	16.1992	0.643351
$635$	26.7483	1.06147
$636$	0	0
$637$	0	0
$638$	−7.24943	−0.287007
$639$	0	0
$640$	−3.20147	−0.126549
$641$	14.3070	0.565094	0.282547	−	0.959253i	$-0.408821\pi$
$641$	14.3070	0.565094	0.282547	+	0.959253i	$0.408821\pi$
$642$	0	0
$643$	−35.4966	−1.39985	−0.699924	−	0.714218i	$-0.746783\pi$
$643$	−35.4966	−1.39985	−0.699924	+	0.714218i	$0.746783\pi$
$644$	0	0
$645$	0	0
$646$	1.15352	0.0453846
$647$	−23.8156	−0.936286	−0.468143	−	0.883653i	$-0.655077\pi$
$647$	−23.8156	−0.936286	−0.468143	+	0.883653i	$0.655077\pi$
$648$	0	0
$649$	11.5565	0.453631
$650$	33.6118	1.31836
$651$	0	0
$652$	13.0553	0.511286
$653$	13.7962	0.539888	0.269944	−	0.962876i	$-0.412995\pi$
$653$	13.7962	0.539888	0.269944	+	0.962876i	$0.412995\pi$
$654$	0	0
$655$	−48.9977	−1.91450
$656$	−8.24943	−0.322086
$657$	0	0
$658$	0	0
$659$	−18.9447	−0.737980	−0.368990	−	0.929433i	$-0.620296\pi$
$659$	−18.9447	−0.737980	−0.368990	+	0.929433i	$0.620296\pi$
$660$	0	0
$661$	3.95941	0.154003	0.0770016	−	0.997031i	$-0.475465\pi$
$661$	3.95941	0.154003	0.0770016	+	0.997031i	$0.475465\pi$
$662$	−21.2471	−0.825793
$663$	0	0
$664$	6.55646	0.254440
$665$	0	0
$666$	0	0
$667$	14.1512	0.547937
$668$	6.70998	0.259617
$669$	0	0
$670$	−20.0074	−0.772952
$671$	−9.70032	−0.374477
$672$	0	0
$673$	−8.69066	−0.335000	−0.167500	−	0.985872i	$-0.553569\pi$
$673$	−8.69066	−0.335000	−0.167500	+	0.985872i	$0.553569\pi$
$674$	25.5159	0.982835
$675$	0	0
$676$	27.9977	1.07683
$677$	31.4606	1.20913	0.604564	−	0.796557i	$-0.293347\pi$
$677$	31.4606	1.20913	0.604564	+	0.796557i	$0.293347\pi$
$678$	0	0
$679$	0	0
$680$	3.20147	0.122771
$681$	0	0
$682$	10.4989	0.402022
$683$	−24.7460	−0.946878	−0.473439	−	0.880826i	$-0.656988\pi$
$683$	−24.7460	−0.946878	−0.473439	+	0.880826i	$0.656988\pi$
$684$	0	0
$685$	20.4989	0.783221
$686$	0	0
$687$	0	0
$688$	5.15352	0.196476
$689$	−40.9977	−1.56189
$690$	0	0
$691$	25.5542	0.972126	0.486063	−	0.873924i	$-0.338433\pi$
$691$	25.5542	0.972126	0.486063	+	0.873924i	$0.338433\pi$
$692$	5.50115	0.209122
$693$	0	0
$694$	3.05531	0.115978
$695$	69.6265	2.64108
$696$	0	0
$697$	8.24943	0.312469
$698$	4.00736	0.151681
$699$	0	0
$700$	0	0
$701$	14.1512	0.534484	0.267242	−	0.963629i	$-0.413888\pi$
$701$	14.1512	0.534484	0.267242	+	0.963629i	$0.413888\pi$
$702$	0	0
$703$	5.94469	0.224208
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−1.09821	−0.0413315
$707$	0	0
$708$	0	0
$709$	−10.7506	−0.403746	−0.201873	−	0.979412i	$-0.564703\pi$
$709$	−10.7506	−0.403746	−0.201873	+	0.979412i	$0.564703\pi$
$710$	16.8059	0.630714
$711$	0	0
$712$	−18.4029	−0.689680
$713$	−20.4943	−0.767516
$714$	0	0
$715$	20.4989	0.766614
$716$	−1.65237	−0.0617520
$717$	0	0
$718$	15.6118	0.582626
$719$	−2.75794	−0.102854	−0.0514268	−	0.998677i	$-0.516377\pi$
$719$	−2.75794	−0.102854	−0.0514268	+	0.998677i	$0.516377\pi$
$720$	0	0
$721$	0	0
$722$	17.6694	0.657587
$723$	0	0
$724$	24.9018	0.925468
$725$	−38.0553	−1.41334
$726$	0	0
$727$	14.6141	0.542006	0.271003	−	0.962578i	$-0.412645\pi$
$727$	14.6141	0.542006	0.271003	+	0.962578i	$0.412645\pi$
$728$	0	0
$729$	0	0
$730$	6.70998	0.248348
$731$	−5.15352	−0.190610
$732$	0	0
$733$	7.10557	0.262450	0.131225	−	0.991353i	$-0.458109\pi$
$733$	7.10557	0.262450	0.131225	+	0.991353i	$0.458109\pi$
$734$	−0.791166	−0.0292025
$735$	0	0
$736$	1.95205	0.0719534
$737$	−6.24943	−0.230201
$738$	0	0
$739$	−41.4966	−1.52648	−0.763238	−	0.646118i	$-0.776391\pi$
$739$	−41.4966	−1.52648	−0.763238	+	0.646118i	$0.776391\pi$
$740$	16.4989	0.606510
$741$	0	0
$742$	0	0
$743$	31.9041	1.17045	0.585224	−	0.810872i	$-0.301007\pi$
$743$	31.9041	1.17045	0.585224	+	0.810872i	$0.301007\pi$
$744$	0	0
$745$	−21.9188	−0.803043
$746$	−15.9115	−0.582560
$747$	0	0
$748$	1.00000	0.0365636
$749$	0	0
$750$	0	0
$751$	13.5159	0.493201	0.246601	−	0.969117i	$-0.420686\pi$
$751$	13.5159	0.493201	0.246601	+	0.969117i	$0.420686\pi$
$752$	6.04795	0.220546
$753$	0	0
$754$	−46.4177	−1.69043
$755$	47.5542	1.73067
$756$	0	0
$757$	−18.5542	−0.674363	−0.337181	−	0.941440i	$-0.609474\pi$
$757$	−18.5542	−0.674363	−0.337181	+	0.941440i	$0.609474\pi$
$758$	17.0553	0.619477
$759$	0	0
$760$	−3.69296	−0.133958
$761$	−15.7506	−0.570958	−0.285479	−	0.958385i	$-0.592153\pi$
$761$	−15.7506	−0.570958	−0.285479	+	0.958385i	$0.592153\pi$
$762$	0	0
$763$	0	0
$764$	4.80589	0.173871
$765$	0	0
$766$	13.4412	0.485652
$767$	73.9954	2.67182
$768$	0	0
$769$	41.2854	1.48879	0.744395	−	0.667739i	$-0.232738\pi$
$769$	41.2854	1.48879	0.744395	+	0.667739i	$0.232738\pi$
$770$	0	0
$771$	0	0
$772$	8.70998	0.313479
$773$	−4.90916	−0.176570	−0.0882850	−	0.996095i	$-0.528139\pi$
$773$	−4.90916	−0.176570	−0.0882850	+	0.996095i	$0.528139\pi$
$774$	0	0
$775$	55.1129	1.97971
$776$	5.49885	0.197397
$777$	0	0
$778$	−7.20147	−0.258185
$779$	−9.51587	−0.340942
$780$	0	0
$781$	5.24943	0.187839
$782$	−1.95205	−0.0698051
$783$	0	0
$784$	0	0
$785$	−30.2877	−1.08101
$786$	0	0
$787$	0.769897	0.0274439	0.0137219	−	0.999906i	$-0.495632\pi$
$787$	0.769897	0.0274439	0.0137219	+	0.999906i	$0.495632\pi$
$788$	−0.654669	−0.0233216
$789$	0	0
$790$	17.9424	0.638361
$791$	0	0
$792$	0	0
$793$	−62.1106	−2.20561
$794$	−12.9424	−0.459308
$795$	0	0
$796$	−7.69296	−0.272670
$797$	−28.9092	−1.02401	−0.512007	−	0.858981i	$-0.671098\pi$
$797$	−28.9092	−1.02401	−0.512007	+	0.858981i	$0.671098\pi$
$798$	0	0
$799$	−6.04795	−0.213961
$800$	−5.24943	−0.185595
$801$	0	0
$802$	−16.9018	−0.596823
$803$	2.09591	0.0739629
$804$	0	0
$805$	0	0
$806$	67.2236	2.36785
$807$	0	0
$808$	−19.9594	−0.702170
$809$	20.2494	0.711932	0.355966	−	0.934499i	$-0.384152\pi$
$809$	20.2494	0.711932	0.355966	+	0.934499i	$0.384152\pi$
$810$	0	0
$811$	−48.4989	−1.70302	−0.851512	−	0.524334i	$-0.824314\pi$
$811$	−48.4989	−1.70302	−0.851512	+	0.524334i	$0.824314\pi$
$812$	0	0
$813$	0	0
$814$	5.15352	0.180631
$815$	41.7962	1.46406
$816$	0	0
$817$	5.94469	0.207978
$818$	37.0936	1.29695
$819$	0	0
$820$	−26.4103	−0.922288
$821$	−31.3047	−1.09254	−0.546271	−	0.837608i	$-0.683953\pi$
$821$	−31.3047	−1.09254	−0.546271	+	0.837608i	$0.683953\pi$
$822$	0	0
$823$	0.287717	0.0100292	0.00501459	−	0.999987i	$-0.498404\pi$
$823$	0.287717	0.0100292	0.00501459	+	0.999987i	$0.498404\pi$
$824$	−0.307039	−0.0106962
$825$	0	0
$826$	0	0
$827$	−37.2471	−1.29521	−0.647605	−	0.761976i	$-0.724229\pi$
$827$	−37.2471	−1.29521	−0.647605	+	0.761976i	$0.724229\pi$
$828$	0	0
$829$	49.1535	1.70717	0.853586	−	0.520952i	$-0.174423\pi$
$829$	49.1535	1.70717	0.853586	+	0.520952i	$0.174423\pi$
$830$	20.9903	0.728585
$831$	0	0
$832$	−6.40294	−0.221982
$833$	0	0
$834$	0	0
$835$	21.4818	0.743409
$836$	−1.15352	−0.0398953
$837$	0	0
$838$	22.4583	0.775808
$839$	2.81785	0.0972830	0.0486415	−	0.998816i	$-0.484511\pi$
$839$	2.81785	0.0972830	0.0486415	+	0.998816i	$0.484511\pi$
$840$	0	0
$841$	23.5542	0.812213
$842$	23.3453	0.804533
$843$	0	0
$844$	14.5948	0.502372
$845$	89.6339	3.08350
$846$	0	0
$847$	0	0
$848$	6.40294	0.219878
$849$	0	0
$850$	5.24943	0.180054
$851$	−10.0599	−0.344849
$852$	0	0
$853$	2.39558	0.0820232	0.0410116	−	0.999159i	$-0.486942\pi$
$853$	2.39558	0.0820232	0.0410116	+	0.999159i	$0.486942\pi$
$854$	0	0
$855$	0	0
$856$	−9.05531	−0.309504
$857$	−0.808189	−0.0276072	−0.0138036	−	0.999905i	$-0.504394\pi$
$857$	−0.808189	−0.0276072	−0.0138036	+	0.999905i	$0.504394\pi$
$858$	0	0
$859$	6.55646	0.223704	0.111852	−	0.993725i	$-0.464322\pi$
$859$	6.55646	0.223704	0.111852	+	0.993725i	$0.464322\pi$
$860$	16.4989	0.562606
$861$	0	0
$862$	31.8229	1.08389
$863$	20.6067	0.701461	0.350730	−	0.936476i	$-0.385933\pi$
$863$	20.6067	0.701461	0.350730	+	0.936476i	$0.385933\pi$
$864$	0	0
$865$	17.6118	0.598818
$866$	−30.3047	−1.02980
$867$	0	0
$868$	0	0
$869$	5.60442	0.190117
$870$	0	0
$871$	−40.0147	−1.35585
$872$	9.70032	0.328494
$873$	0	0
$874$	2.25172	0.0761657
$875$	0	0
$876$	0	0
$877$	26.6021	0.898290	0.449145	−	0.893459i	$-0.351729\pi$
$877$	26.6021	0.898290	0.449145	+	0.893459i	$0.351729\pi$
$878$	10.6620	0.359826
$879$	0	0
$880$	−3.20147	−0.107922
$881$	8.49885	0.286334	0.143167	−	0.989699i	$-0.454271\pi$
$881$	8.49885	0.286334	0.143167	+	0.989699i	$0.454271\pi$
$882$	0	0
$883$	46.2448	1.55626	0.778131	−	0.628102i	$-0.216168\pi$
$883$	46.2448	1.55626	0.778131	+	0.628102i	$0.216168\pi$
$884$	6.40294	0.215354
$885$	0	0
$886$	−36.0553	−1.21130
$887$	−18.2877	−0.614041	−0.307021	−	0.951703i	$-0.599332\pi$
$887$	−18.2877	−0.614041	−0.307021	+	0.951703i	$0.599332\pi$
$888$	0	0
$889$	0	0
$890$	−58.9165	−1.97489
$891$	0	0
$892$	−14.9018	−0.498949
$893$	6.97643	0.233457
$894$	0	0
$895$	−5.29002	−0.176826
$896$	0	0
$897$	0	0
$898$	23.4006	0.780890
$899$	−76.1106	−2.53843
$900$	0	0
$901$	−6.40294	−0.213313
$902$	−8.24943	−0.274676
$903$	0	0
$904$	−12.4989	−0.415706
$905$	79.7224	2.65006
$906$	0	0
$907$	−31.6694	−1.05156	−0.525782	−	0.850619i	$-0.676227\pi$
$907$	−31.6694	−1.05156	−0.525782	+	0.850619i	$0.676227\pi$
$908$	0.249425	0.00827746
$909$	0	0
$910$	0	0
$911$	−58.7727	−1.94723	−0.973613	−	0.228207i	$-0.926714\pi$
$911$	−58.7727	−1.94723	−0.973613	+	0.228207i	$0.926714\pi$
$912$	0	0
$913$	6.55646	0.216987
$914$	18.6141	0.615699
$915$	0	0
$916$	−4.09591	−0.135333
$917$	0	0
$918$	0	0
$919$	−32.9691	−1.08755	−0.543775	−	0.839231i	$-0.683005\pi$
$919$	−32.9691	−1.08755	−0.543775	+	0.839231i	$0.683005\pi$
$920$	6.24943	0.206037
$921$	0	0
$922$	−31.4412	−1.03546
$923$	33.6118	1.10635
$924$	0	0
$925$	27.0530	0.889498
$926$	22.0959	0.726117
$927$	0	0
$928$	7.24943	0.237974
$929$	−41.8995	−1.37468	−0.687339	−	0.726337i	$-0.741221\pi$
$929$	−41.8995	−1.37468	−0.687339	+	0.726337i	$0.741221\pi$
$930$	0	0
$931$	0	0
$932$	−11.3070	−0.370374
$933$	0	0
$934$	21.0576	0.689026
$935$	3.20147	0.104699
$936$	0	0
$937$	−33.4966	−1.09428	−0.547142	−	0.837040i	$-0.684284\pi$
$937$	−33.4966	−1.09428	−0.547142	+	0.837040i	$0.684284\pi$
$938$	0	0
$939$	0	0
$940$	19.3624	0.631530
$941$	45.7483	1.49135	0.745676	−	0.666309i	$-0.232127\pi$
$941$	45.7483	1.49135	0.745676	+	0.666309i	$0.232127\pi$
$942$	0	0
$943$	16.1033	0.524395
$944$	−11.5565	−0.376131
$945$	0	0
$946$	5.15352	0.167555
$947$	45.0530	1.46403	0.732013	−	0.681291i	$-0.238581\pi$
$947$	45.0530	1.46403	0.732013	+	0.681291i	$0.238581\pi$
$948$	0	0
$949$	13.4200	0.435631
$950$	−6.05531	−0.196460
$951$	0	0
$952$	0	0
$953$	−37.1705	−1.20407	−0.602036	−	0.798469i	$-0.705644\pi$
$953$	−37.1705	−1.20407	−0.602036	+	0.798469i	$0.705644\pi$
$954$	0	0
$955$	15.3859	0.497877
$956$	−6.59476	−0.213290
$957$	0	0
$958$	−21.2088	−0.685226
$959$	0	0
$960$	0	0
$961$	79.2259	2.55567
$962$	32.9977	1.06389
$963$	0	0
$964$	17.1129	0.551170
$965$	27.8848	0.897643
$966$	0	0
$967$	42.7579	1.37500	0.687501	−	0.726183i	$-0.258708\pi$
$967$	42.7579	1.37500	0.687501	+	0.726183i	$0.258708\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	17.6044	0.565244
$971$	−0.690661	−0.0221644	−0.0110822	−	0.999939i	$-0.503528\pi$
$971$	−0.690661	−0.0221644	−0.0110822	+	0.999939i	$0.503528\pi$
$972$	0	0
$973$	0	0
$974$	38.7100	1.24035
$975$	0	0
$976$	9.70032	0.310500
$977$	33.1895	1.06183	0.530913	−	0.847426i	$-0.321849\pi$
$977$	33.1895	1.06183	0.530913	+	0.847426i	$0.321849\pi$
$978$	0	0
$979$	−18.4029	−0.588161
$980$	0	0
$981$	0	0
$982$	19.1705	0.611757
$983$	42.6620	1.36071	0.680354	−	0.732884i	$-0.261826\pi$
$983$	42.6620	1.36071	0.680354	+	0.732884i	$0.261826\pi$
$984$	0	0
$985$	−2.09591	−0.0667811
$986$	−7.24943	−0.230869
$987$	0	0
$988$	−7.38592	−0.234977
$989$	−10.0599	−0.319887
$990$	0	0
$991$	−24.0147	−0.762853	−0.381426	−	0.924399i	$-0.624567\pi$
$991$	−24.0147	−0.762853	−0.381426	+	0.924399i	$0.624567\pi$
$992$	−10.4989	−0.333339
$993$	0	0
$994$	0	0
$995$	−24.6288	−0.780785
$996$	0	0
$997$	33.5971	1.06403	0.532015	−	0.846735i	$-0.321435\pi$
$997$	33.5971	1.06403	0.532015	+	0.846735i	$0.321435\pi$
$998$	27.9188	0.883755
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dt.1.3		3
3.2	odd	2		3234.2.a.bi.1.1		3
7.2	even	3		1386.2.k.w.991.1		6
7.4	even	3		1386.2.k.w.793.1		6
7.6	odd	2		9702.2.a.du.1.1		3
21.2	odd	6		462.2.i.f.67.3	✓	6
21.11	odd	6		462.2.i.f.331.3	yes	6
21.20	even	2		3234.2.a.bg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.f.67.3	✓	6	21.2	odd	6
462.2.i.f.331.3	yes	6	21.11	odd	6
1386.2.k.w.793.1		6	7.4	even	3
1386.2.k.w.991.1		6	7.2	even	3
3234.2.a.bg.1.3		3	21.20	even	2
3234.2.a.bi.1.1		3	3.2	odd	2
9702.2.a.dt.1.3		3	1.1	even	1	trivial
9702.2.a.du.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.20147 q^{5} -1.00000 q^{8} -3.20147 q^{10} -1.00000 q^{11} -6.40294 q^{13} +1.00000 q^{16} -1.00000 q^{17} +1.15352 q^{19} +3.20147 q^{20} +1.00000 q^{22} -1.95205 q^{23} +5.24943 q^{25} +6.40294 q^{26} -7.24943 q^{29} +10.4989 q^{31} -1.00000 q^{32} +1.00000 q^{34} +5.15352 q^{37} -1.15352 q^{38} -3.20147 q^{40} -8.24943 q^{41} +5.15352 q^{43} -1.00000 q^{44} +1.95205 q^{46} +6.04795 q^{47} -5.24943 q^{50} -6.40294 q^{52} +6.40294 q^{53} -3.20147 q^{55} +7.24943 q^{58} -11.5565 q^{59} +9.70032 q^{61} -10.4989 q^{62} +1.00000 q^{64} -20.4989 q^{65} +6.24943 q^{67} -1.00000 q^{68} -5.24943 q^{71} -2.09591 q^{73} -5.15352 q^{74} +1.15352 q^{76} -5.60442 q^{79} +3.20147 q^{80} +8.24943 q^{82} -6.55646 q^{83} -3.20147 q^{85} -5.15352 q^{86} +1.00000 q^{88} +18.4029 q^{89} -1.95205 q^{92} -6.04795 q^{94} +3.69296 q^{95} -5.49885 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 3 q^{11} + 3 q^{16} - 3 q^{17} - 3 q^{19} + 3 q^{22} - 9 q^{23} + 3 q^{25} - 9 q^{29} + 6 q^{31} - 3 q^{32} + 3 q^{34} + 9 q^{37} + 3 q^{38} - 12 q^{41} + 9 q^{43} - 3 q^{44} + 9 q^{46} + 15 q^{47} - 3 q^{50} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} - 36 q^{65} + 6 q^{67} - 3 q^{68} - 3 q^{71} - 9 q^{74} - 3 q^{76} + 12 q^{79} + 12 q^{82} + 6 q^{83} - 9 q^{86} + 3 q^{88} + 36 q^{89} - 9 q^{92} - 15 q^{94} + 24 q^{95} + 9 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 - 3 * q^11 + 3 * q^16 - 3 * q^17 - 3 * q^19 + 3 * q^22 - 9 * q^23 + 3 * q^25 - 9 * q^29 + 6 * q^31 - 3 * q^32 + 3 * q^34 + 9 * q^37 + 3 * q^38 - 12 * q^41 + 9 * q^43 - 3 * q^44 + 9 * q^46 + 15 * q^47 - 3 * q^50 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 - 36 * q^65 + 6 * q^67 - 3 * q^68 - 3 * q^71 - 9 * q^74 - 3 * q^76 + 12 * q^79 + 12 * q^82 + 6 * q^83 - 9 * q^86 + 3 * q^88 + 36 * q^89 - 9 * q^92 - 15 * q^94 + 24 * q^95 + 9 * q^97

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