LMFDB - Embedded newform 9747.2.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9747,2,Mod(1,9747)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9747, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9747.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9747 = 3^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9747.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,-3,0,-9,-6,0,0,-9,0,-9,12,0,3,-3,0,0,-3,0,18,3,0,-6, 12,0,-18,-6,0,0,0,0,-9,12,0,-27,0,0,18,-21,0,-18,-18,0,-9,-6,0,12,15,0, -9,-6,0,0,33,0,-9,-3,0,-9,3,0,12,3,0,-9,33,0,-9,-3,0,-18,21,0,0,27,0,0, -12,0,18,0,0,-9,3,0,9,-33,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$77.8301868501$
Analytic rank:	$2$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 513)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	9747.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.879385 q^{2} -1.22668 q^{4} -2.53209 q^{5} -2.65270 q^{7} -2.83750 q^{8} -2.22668 q^{10} +0.411474 q^{11} -1.46791 q^{13} -2.33275 q^{14} -0.0418891 q^{16} -2.38919 q^{17} +3.10607 q^{20} +0.361844 q^{22} -0.879385 q^{23} +1.41147 q^{25} -1.29086 q^{26} +3.25402 q^{28} -6.10607 q^{29} -2.02229 q^{31} +5.63816 q^{32} -2.10101 q^{34} +6.71688 q^{35} -9.69459 q^{37} +7.18479 q^{40} -8.53209 q^{41} -11.2909 q^{43} -0.504748 q^{44} -0.773318 q^{46} -8.47565 q^{47} +0.0368366 q^{49} +1.24123 q^{50} +1.80066 q^{52} -1.50980 q^{53} -1.04189 q^{55} +7.52704 q^{56} -5.36959 q^{58} -3.71688 q^{59} -2.50980 q^{61} -1.77837 q^{62} +5.04189 q^{64} +3.71688 q^{65} -4.87939 q^{67} +2.93077 q^{68} +5.90673 q^{70} +10.0719 q^{71} -10.2490 q^{73} -8.52528 q^{74} -1.09152 q^{77} +16.2199 q^{79} +0.106067 q^{80} -7.50299 q^{82} -5.63816 q^{83} +6.04963 q^{85} -9.92902 q^{86} -1.16756 q^{88} -1.60307 q^{89} +3.89393 q^{91} +1.07873 q^{92} -7.45336 q^{94} -0.101014 q^{97} +0.0323936 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 9 q^{7} - 6 q^{8} - 9 q^{11} - 9 q^{13} + 12 q^{14} + 3 q^{16} - 3 q^{17} - 3 q^{20} + 18 q^{22} + 3 q^{23} - 6 q^{25} + 12 q^{26} - 18 q^{28} - 6 q^{29} - 9 q^{34} + 12 q^{35}+ \cdots - 36 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 9 * q^7 - 6 * q^8 - 9 * q^11 - 9 * q^13 + 12 * q^14 + 3 * q^16 - 3 * q^17 - 3 * q^20 + 18 * q^22 + 3 * q^23 - 6 * q^25 + 12 * q^26 - 18 * q^28 - 6 * q^29 - 9 * q^34 + 12 * q^35 - 27 * q^37 + 18 * q^40 - 21 * q^41 - 18 * q^43 - 18 * q^44 - 9 * q^46 - 6 * q^47 + 12 * q^49 + 15 * q^50 - 9 * q^52 - 6 * q^53 + 33 * q^56 - 9 * q^58 - 3 * q^59 - 9 * q^61 + 3 * q^62 + 12 * q^64 + 3 * q^65 - 9 * q^67 + 33 * q^68 - 9 * q^70 - 3 * q^71 - 18 * q^73 + 21 * q^74 + 27 * q^77 - 12 * q^80 + 18 * q^82 - 9 * q^85 + 3 * q^86 + 9 * q^88 - 33 * q^89 + 24 * q^91 + 12 * q^92 - 9 * q^94 - 3 * q^97 - 36 * q^98

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 $

$2$	0.879385	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$2$	0.879385	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$3$	0	0
$3$	0	0
$4$	−1.22668	−0.613341
$5$	−2.53209	−1.13238	−0.566192	−	0.824273i	$-0.691584\pi$
$5$	−2.53209	−1.13238	−0.566192	+	0.824273i	$0.691584\pi$
$6$	0	0
$7$	−2.65270	−1.00263	−0.501314	−	0.865266i	$-0.667150\pi$
$7$	−2.65270	−1.00263	−0.501314	+	0.865266i	$0.667150\pi$
$8$	−2.83750	−1.00321
$9$	0	0
$10$	−2.22668	−0.704139
$11$	0.411474	0.124064	0.0620321	−	0.998074i	$-0.480242\pi$
$11$	0.411474	0.124064	0.0620321	+	0.998074i	$0.480242\pi$
$12$	0	0
$13$	−1.46791	−0.407125	−0.203563	−	0.979062i	$-0.565252\pi$
$13$	−1.46791	−0.407125	−0.203563	+	0.979062i	$0.565252\pi$
$14$	−2.33275	−0.623453
$15$	0	0
$16$	−0.0418891	−0.0104723
$17$	−2.38919	−0.579463	−0.289731	−	0.957108i	$-0.593566\pi$
$17$	−2.38919	−0.579463	−0.289731	+	0.957108i	$0.593566\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.10607	0.694538
$21$	0	0
$22$	0.361844	0.0771455
$23$	−0.879385	−0.183364	−0.0916822	−	0.995788i	$-0.529224\pi$
$23$	−0.879385	−0.183364	−0.0916822	+	0.995788i	$0.529224\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	−1.29086	−0.253158
$27$	0	0
$28$	3.25402	0.614952
$29$	−6.10607	−1.13387	−0.566934	−	0.823763i	$-0.691871\pi$
$29$	−6.10607	−1.13387	−0.566934	+	0.823763i	$0.691871\pi$
$30$	0	0
$31$	−2.02229	−0.363214	−0.181607	−	0.983371i	$-0.558130\pi$
$31$	−2.02229	−0.363214	−0.181607	+	0.983371i	$0.558130\pi$
$32$	5.63816	0.996695
$33$	0	0
$34$	−2.10101	−0.360321
$35$	6.71688	1.13536
$36$	0	0
$37$	−9.69459	−1.59378	−0.796891	−	0.604124i	$-0.793523\pi$
$37$	−9.69459	−1.59378	−0.796891	+	0.604124i	$0.793523\pi$
$38$	0	0
$39$	0	0
$40$	7.18479	1.13602
$41$	−8.53209	−1.33249	−0.666244	−	0.745734i	$-0.732099\pi$
$41$	−8.53209	−1.33249	−0.666244	+	0.745734i	$0.732099\pi$
$42$	0	0
$43$	−11.2909	−1.72184	−0.860920	−	0.508740i	$-0.830111\pi$
$43$	−11.2909	−1.72184	−0.860920	+	0.508740i	$0.830111\pi$
$44$	−0.504748	−0.0760936
$45$	0	0
$46$	−0.773318	−0.114020
$47$	−8.47565	−1.23630	−0.618150	−	0.786060i	$-0.712118\pi$
$47$	−8.47565	−1.23630	−0.618150	+	0.786060i	$0.712118\pi$
$48$	0	0
$49$	0.0368366	0.00526238
$50$	1.24123	0.175536
$51$	0	0
$52$	1.80066	0.249707
$53$	−1.50980	−0.207387	−0.103694	−	0.994609i	$-0.533066\pi$
$53$	−1.50980	−0.207387	−0.103694	+	0.994609i	$0.533066\pi$
$54$	0	0
$55$	−1.04189	−0.140488
$56$	7.52704	1.00584
$57$	0	0
$58$	−5.36959	−0.705061
$59$	−3.71688	−0.483897	−0.241948	−	0.970289i	$-0.577786\pi$
$59$	−3.71688	−0.483897	−0.241948	+	0.970289i	$0.577786\pi$
$60$	0	0
$61$	−2.50980	−0.321347	−0.160673	−	0.987008i	$-0.551367\pi$
$61$	−2.50980	−0.321347	−0.160673	+	0.987008i	$0.551367\pi$
$62$	−1.77837	−0.225853
$63$	0	0
$64$	5.04189	0.630236
$65$	3.71688	0.461022
$66$	0	0
$67$	−4.87939	−0.596112	−0.298056	−	0.954548i	$-0.596338\pi$
$67$	−4.87939	−0.596112	−0.298056	+	0.954548i	$0.596338\pi$
$68$	2.93077	0.355408
$69$	0	0
$70$	5.90673	0.705989
$71$	10.0719	1.19532	0.597658	−	0.801751i	$-0.296098\pi$
$71$	10.0719	1.19532	0.597658	+	0.801751i	$0.296098\pi$
$72$	0	0
$73$	−10.2490	−1.19955	−0.599776	−	0.800168i	$-0.704743\pi$
$73$	−10.2490	−1.19955	−0.599776	+	0.800168i	$0.704743\pi$
$74$	−8.52528	−0.991044
$75$	0	0
$76$	0	0
$77$	−1.09152	−0.124390
$78$	0	0
$79$	16.2199	1.82488	0.912439	−	0.409212i	$-0.134196\pi$
$79$	16.2199	1.82488	0.912439	+	0.409212i	$0.134196\pi$
$80$	0.106067	0.0118586
$81$	0	0
$82$	−7.50299	−0.828567
$83$	−5.63816	−0.618868	−0.309434	−	0.950921i	$-0.600140\pi$
$83$	−5.63816	−0.618868	−0.309434	+	0.950921i	$0.600140\pi$
$84$	0	0
$85$	6.04963	0.656174
$86$	−9.92902	−1.07067
$87$	0	0
$88$	−1.16756	−0.124462
$89$	−1.60307	−0.169925	−0.0849627	−	0.996384i	$-0.527077\pi$
$89$	−1.60307	−0.169925	−0.0849627	+	0.996384i	$0.527077\pi$
$90$	0	0
$91$	3.89393	0.408195
$92$	1.07873	0.112465
$93$	0	0
$94$	−7.45336	−0.768756
$95$	0	0
$96$	0	0
$97$	−0.101014	−0.0102565	−0.00512823	−	0.999987i	$-0.501632\pi$
$97$	−0.101014	−0.0102565	−0.00512823	+	0.999987i	$0.501632\pi$
$98$	0.0323936	0.00327225
$99$	0	0
$100$	−1.73143	−0.173143
$101$	−10.4338	−1.03820	−0.519099	−	0.854714i	$-0.673732\pi$
$101$	−10.4338	−1.03820	−0.519099	+	0.854714i	$0.673732\pi$
$102$	0	0
$103$	7.17024	0.706505	0.353253	−	0.935528i	$-0.385076\pi$
$103$	7.17024	0.706505	0.353253	+	0.935528i	$0.385076\pi$
$104$	4.16519	0.408431
$105$	0	0
$106$	−1.32770	−0.128957
$107$	12.8726	1.24444	0.622220	−	0.782843i	$-0.286231\pi$
$107$	12.8726	1.24444	0.622220	+	0.782843i	$0.286231\pi$
$108$	0	0
$109$	−12.7811	−1.22420	−0.612102	−	0.790779i	$-0.709676\pi$
$109$	−12.7811	−1.22420	−0.612102	+	0.790779i	$0.709676\pi$
$110$	−0.916222	−0.0873583
$111$	0	0
$112$	0.111119	0.0104998
$113$	−11.0077	−1.03552	−0.517761	−	0.855526i	$-0.673234\pi$
$113$	−11.0077	−1.03552	−0.517761	+	0.855526i	$0.673234\pi$
$114$	0	0
$115$	2.22668	0.207639
$116$	7.49020	0.695448
$117$	0	0
$118$	−3.26857	−0.300896
$119$	6.33780	0.580985
$120$	0	0
$121$	−10.8307	−0.984608
$122$	−2.20708	−0.199820
$123$	0	0
$124$	2.48070	0.222774
$125$	9.08647	0.812718
$126$	0	0
$127$	−5.54664	−0.492184	−0.246092	−	0.969246i	$-0.579147\pi$
$127$	−5.54664	−0.492184	−0.246092	+	0.969246i	$0.579147\pi$
$128$	−6.84255	−0.604802
$129$	0	0
$130$	3.26857	0.286673
$131$	−19.5895	−1.71154	−0.855770	−	0.517357i	$-0.826916\pi$
$131$	−19.5895	−1.71154	−0.855770	+	0.517357i	$0.826916\pi$
$132$	0	0
$133$	0	0
$134$	−4.29086	−0.370674
$135$	0	0
$136$	6.77930	0.581321
$137$	19.3037	1.64922	0.824611	−	0.565700i	$-0.191394\pi$
$137$	19.3037	1.64922	0.824611	+	0.565700i	$0.191394\pi$
$138$	0	0
$139$	3.17799	0.269553	0.134777	−	0.990876i	$-0.456968\pi$
$139$	3.17799	0.269553	0.134777	+	0.990876i	$0.456968\pi$
$140$	−8.23947	−0.696363
$141$	0	0
$142$	8.85710	0.743271
$143$	−0.604007	−0.0505096
$144$	0	0
$145$	15.4611	1.28397
$146$	−9.01279	−0.745904
$147$	0	0
$148$	11.8922	0.977531
$149$	10.5098	0.860996	0.430498	−	0.902591i	$-0.358338\pi$
$149$	10.5098	0.860996	0.430498	+	0.902591i	$0.358338\pi$
$150$	0	0
$151$	9.59627	0.780933	0.390467	−	0.920617i	$-0.372314\pi$
$151$	9.59627	0.780933	0.390467	+	0.920617i	$0.372314\pi$
$152$	0	0
$153$	0	0
$154$	−0.959866	−0.0773482
$155$	5.12061	0.411298
$156$	0	0
$157$	−15.3824	−1.22765	−0.613824	−	0.789443i	$-0.710369\pi$
$157$	−15.3824	−1.22765	−0.613824	+	0.789443i	$0.710369\pi$
$158$	14.2635	1.13474
$159$	0	0
$160$	−14.2763	−1.12864
$161$	2.33275	0.183846
$162$	0	0
$163$	18.3969	1.44096	0.720479	−	0.693477i	$-0.243922\pi$
$163$	18.3969	1.44096	0.720479	+	0.693477i	$0.243922\pi$
$164$	10.4662	0.817270
$165$	0	0
$166$	−4.95811	−0.384824
$167$	−22.8949	−1.77166	−0.885829	−	0.464012i	$-0.846410\pi$
$167$	−22.8949	−1.77166	−0.885829	+	0.464012i	$0.846410\pi$
$168$	0	0
$169$	−10.8452	−0.834249
$170$	5.31996	0.408022
$171$	0	0
$172$	13.8503	1.05607
$173$	10.8280	0.823238	0.411619	−	0.911356i	$-0.364964\pi$
$173$	10.8280	0.823238	0.411619	+	0.911356i	$0.364964\pi$
$174$	0	0
$175$	−3.74422	−0.283037
$176$	−0.0172363	−0.00129923
$177$	0	0
$178$	−1.40972	−0.105663
$179$	−6.55438	−0.489897	−0.244949	−	0.969536i	$-0.578771\pi$
$179$	−6.55438	−0.489897	−0.244949	+	0.969536i	$0.578771\pi$
$180$	0	0
$181$	5.60401	0.416543	0.208271	−	0.978071i	$-0.433216\pi$
$181$	5.60401	0.416543	0.208271	+	0.978071i	$0.433216\pi$
$182$	3.42427	0.253824
$183$	0	0
$184$	2.49525	0.183952
$185$	24.5476	1.80477
$186$	0	0
$187$	−0.983088	−0.0718905
$188$	10.3969	0.758274
$189$	0	0
$190$	0	0
$191$	3.26857	0.236505	0.118253	−	0.992984i	$-0.462271\pi$
$191$	3.26857	0.236505	0.118253	+	0.992984i	$0.462271\pi$
$192$	0	0
$193$	−1.59358	−0.114708	−0.0573541	−	0.998354i	$-0.518266\pi$
$193$	−1.59358	−0.114708	−0.0573541	+	0.998354i	$0.518266\pi$
$194$	−0.0888306	−0.00637766
$195$	0	0
$196$	−0.0451868	−0.00322763
$197$	21.2686	1.51532	0.757661	−	0.652648i	$-0.226342\pi$
$197$	21.2686	1.51532	0.757661	+	0.652648i	$0.226342\pi$
$198$	0	0
$199$	−17.8280	−1.26379	−0.631897	−	0.775053i	$-0.717723\pi$
$199$	−17.8280	−1.26379	−0.631897	+	0.775053i	$0.717723\pi$
$200$	−4.00505	−0.283200
$201$	0	0
$202$	−9.17530	−0.645572
$203$	16.1976	1.13685
$204$	0	0
$205$	21.6040	1.50889
$206$	6.30541	0.439319
$207$	0	0
$208$	0.0614894	0.00426352
$209$	0	0
$210$	0	0
$211$	21.0847	1.45153	0.725766	−	0.687942i	$-0.241486\pi$
$211$	21.0847	1.45153	0.725766	+	0.687942i	$0.241486\pi$
$212$	1.85204	0.127199
$213$	0	0
$214$	11.3200	0.773816
$215$	28.5895	1.94978
$216$	0	0
$217$	5.36453	0.364168
$218$	−11.2395	−0.761233
$219$	0	0
$220$	1.27807	0.0861672
$221$	3.50711	0.235914
$222$	0	0
$223$	19.7246	1.32086	0.660429	−	0.750888i	$-0.270374\pi$
$223$	19.7246	1.32086	0.660429	+	0.750888i	$0.270374\pi$
$224$	−14.9564	−0.999314
$225$	0	0
$226$	−9.68004	−0.643907
$227$	26.4020	1.75236	0.876180	−	0.481983i	$-0.160083\pi$
$227$	26.4020	1.75236	0.876180	+	0.481983i	$0.160083\pi$
$228$	0	0
$229$	−5.87164	−0.388009	−0.194005	−	0.981001i	$-0.562148\pi$
$229$	−5.87164	−0.388009	−0.194005	+	0.981001i	$0.562148\pi$
$230$	1.95811	0.129114
$231$	0	0
$232$	17.3259	1.13750
$233$	25.9881	1.70254	0.851270	−	0.524728i	$-0.175833\pi$
$233$	25.9881	1.70254	0.851270	+	0.524728i	$0.175833\pi$
$234$	0	0
$235$	21.4611	1.39997
$236$	4.55943	0.296794
$237$	0	0
$238$	5.57337	0.361268
$239$	−8.36959	−0.541383	−0.270692	−	0.962666i	$-0.587252\pi$
$239$	−8.36959	−0.541383	−0.270692	+	0.962666i	$0.587252\pi$
$240$	0	0
$241$	−10.8794	−0.700803	−0.350401	−	0.936600i	$-0.613955\pi$
$241$	−10.8794	−0.700803	−0.350401	+	0.936600i	$0.613955\pi$
$242$	−9.52435	−0.612248
$243$	0	0
$244$	3.07873	0.197095
$245$	−0.0932736	−0.00595903
$246$	0	0
$247$	0	0
$248$	5.73824	0.364378
$249$	0	0
$250$	7.99050	0.505364
$251$	−6.19934	−0.391299	−0.195649	−	0.980674i	$-0.562681\pi$
$251$	−6.19934	−0.391299	−0.195649	+	0.980674i	$0.562681\pi$
$252$	0	0
$253$	−0.361844	−0.0227490
$254$	−4.87763	−0.306050
$255$	0	0
$256$	−16.1010	−1.00631
$257$	−16.9213	−1.05552	−0.527760	−	0.849394i	$-0.676968\pi$
$257$	−16.9213	−1.05552	−0.527760	+	0.849394i	$0.676968\pi$
$258$	0	0
$259$	25.7169	1.59797
$260$	−4.55943	−0.282764
$261$	0	0
$262$	−17.2267	−1.06427
$263$	14.6013	0.900356	0.450178	−	0.892939i	$-0.351361\pi$
$263$	14.6013	0.900356	0.450178	+	0.892939i	$0.351361\pi$
$264$	0	0
$265$	3.82295	0.234842
$266$	0	0
$267$	0	0
$268$	5.98545	0.365620
$269$	17.2567	1.05216	0.526080	−	0.850435i	$-0.323661\pi$
$269$	17.2567	1.05216	0.526080	+	0.850435i	$0.323661\pi$
$270$	0	0
$271$	−4.63041	−0.281278	−0.140639	−	0.990061i	$-0.544916\pi$
$271$	−4.63041	−0.281278	−0.140639	+	0.990061i	$0.544916\pi$
$272$	0.100081	0.00606829
$273$	0	0
$274$	16.9753	1.02552
$275$	0.580785	0.0350227
$276$	0	0
$277$	−4.72369	−0.283819	−0.141909	−	0.989880i	$-0.545324\pi$
$277$	−4.72369	−0.283819	−0.141909	+	0.989880i	$0.545324\pi$
$278$	2.79467	0.167613
$279$	0	0
$280$	−19.0591	−1.13900
$281$	−32.6509	−1.94779	−0.973896	−	0.226994i	$-0.927110\pi$
$281$	−32.6509	−1.94779	−0.973896	+	0.226994i	$0.927110\pi$
$282$	0	0
$283$	−9.78106	−0.581424	−0.290712	−	0.956811i	$-0.593892\pi$
$283$	−9.78106	−0.581424	−0.290712	+	0.956811i	$0.593892\pi$
$284$	−12.3550	−0.733137
$285$	0	0
$286$	−0.531155	−0.0314079
$287$	22.6331	1.33599
$288$	0	0
$289$	−11.2918	−0.664223
$290$	13.5963	0.798400
$291$	0	0
$292$	12.5722	0.735734
$293$	4.13516	0.241579	0.120789	−	0.992678i	$-0.461457\pi$
$293$	4.13516	0.241579	0.120789	+	0.992678i	$0.461457\pi$
$294$	0	0
$295$	9.41147	0.547957
$296$	27.5084	1.59889
$297$	0	0
$298$	9.24216	0.535384
$299$	1.29086	0.0746523
$300$	0	0
$301$	29.9513	1.72636
$302$	8.43882	0.485599
$303$	0	0
$304$	0	0
$305$	6.35504	0.363888
$306$	0	0
$307$	−6.22668	−0.355375	−0.177688	−	0.984087i	$-0.556862\pi$
$307$	−6.22668	−0.355375	−0.177688	+	0.984087i	$0.556862\pi$
$308$	1.33895	0.0762935
$309$	0	0
$310$	4.50299	0.255753
$311$	−2.58853	−0.146782	−0.0733909	−	0.997303i	$-0.523382\pi$
$311$	−2.58853	−0.146782	−0.0733909	+	0.997303i	$0.523382\pi$
$312$	0	0
$313$	−20.6723	−1.16847	−0.584234	−	0.811585i	$-0.698605\pi$
$313$	−20.6723	−1.16847	−0.584234	+	0.811585i	$0.698605\pi$
$314$	−13.5270	−0.763375
$315$	0	0
$316$	−19.8966	−1.11927
$317$	9.13610	0.513134	0.256567	−	0.966526i	$-0.417409\pi$
$317$	9.13610	0.513134	0.256567	+	0.966526i	$0.417409\pi$
$318$	0	0
$319$	−2.51249	−0.140672
$320$	−12.7665	−0.713670
$321$	0	0
$322$	2.05138	0.114319
$323$	0	0
$324$	0	0
$325$	−2.07192	−0.114929
$326$	16.1780	0.896016
$327$	0	0
$328$	24.2098	1.33676
$329$	22.4834	1.23955
$330$	0	0
$331$	16.0942	0.884618	0.442309	−	0.896863i	$-0.354160\pi$
$331$	16.0942	0.884618	0.442309	+	0.896863i	$0.354160\pi$
$332$	6.91622	0.379577
$333$	0	0
$334$	−20.1334	−1.10165
$335$	12.3550	0.675028
$336$	0	0
$337$	−20.4397	−1.11342	−0.556712	−	0.830706i	$-0.687937\pi$
$337$	−20.4397	−1.11342	−0.556712	+	0.830706i	$0.687937\pi$
$338$	−9.53714	−0.518752
$339$	0	0
$340$	−7.42097	−0.402459
$341$	−0.832119	−0.0450618
$342$	0	0
$343$	18.4712	0.997352
$344$	32.0378	1.72736
$345$	0	0
$346$	9.52198	0.511905
$347$	−22.0847	−1.18557	−0.592785	−	0.805361i	$-0.701972\pi$
$347$	−22.0847	−1.18557	−0.592785	+	0.805361i	$0.701972\pi$
$348$	0	0
$349$	−18.5253	−0.991636	−0.495818	−	0.868427i	$-0.665132\pi$
$349$	−18.5253	−0.991636	−0.495818	+	0.868427i	$0.665132\pi$
$350$	−3.29261	−0.175998
$351$	0	0
$352$	2.31996	0.123654
$353$	31.1088	1.65575	0.827876	−	0.560911i	$-0.189549\pi$
$353$	31.1088	1.65575	0.827876	+	0.560911i	$0.189549\pi$
$354$	0	0
$355$	−25.5030	−1.35356
$356$	1.96646	0.104222
$357$	0	0
$358$	−5.76382	−0.304628
$359$	−17.0009	−0.897275	−0.448637	−	0.893714i	$-0.648091\pi$
$359$	−17.0009	−0.897275	−0.448637	+	0.893714i	$0.648091\pi$
$360$	0	0
$361$	0	0
$362$	4.92808	0.259014
$363$	0	0
$364$	−4.77662	−0.250363
$365$	25.9513	1.35835
$366$	0	0
$367$	22.9932	1.20023	0.600117	−	0.799912i	$-0.295121\pi$
$367$	22.9932	1.20023	0.600117	+	0.799912i	$0.295121\pi$
$368$	0.0368366	0.00192024
$369$	0	0
$370$	21.5868	1.12224
$371$	4.00505	0.207932
$372$	0	0
$373$	3.42696	0.177441	0.0887205	−	0.996057i	$-0.471722\pi$
$373$	3.42696	0.177441	0.0887205	+	0.996057i	$0.471722\pi$
$374$	−0.864513	−0.0447029
$375$	0	0
$376$	24.0496	1.24026
$377$	8.96316	0.461626
$378$	0	0
$379$	−12.1898	−0.626150	−0.313075	−	0.949728i	$-0.601359\pi$
$379$	−12.1898	−0.626150	−0.313075	+	0.949728i	$0.601359\pi$
$380$	0	0
$381$	0	0
$382$	2.87433	0.147064
$383$	−9.26176	−0.473254	−0.236627	−	0.971601i	$-0.576042\pi$
$383$	−9.26176	−0.473254	−0.236627	+	0.971601i	$0.576042\pi$
$384$	0	0
$385$	2.76382	0.140857
$386$	−1.40137	−0.0713278
$387$	0	0
$388$	0.123913	0.00629070
$389$	−36.4739	−1.84930	−0.924650	−	0.380818i	$-0.875642\pi$
$389$	−36.4739	−1.84930	−0.924650	+	0.380818i	$0.875642\pi$
$390$	0	0
$391$	2.10101	0.106253
$392$	−0.104524	−0.00527925
$393$	0	0
$394$	18.7033	0.942257
$395$	−41.0702	−2.06646
$396$	0	0
$397$	−15.6682	−0.786364	−0.393182	−	0.919461i	$-0.628626\pi$
$397$	−15.6682	−0.786364	−0.393182	+	0.919461i	$0.628626\pi$
$398$	−15.6777	−0.785851
$399$	0	0
$400$	−0.0591253	−0.00295627
$401$	−9.23854	−0.461351	−0.230675	−	0.973031i	$-0.574094\pi$
$401$	−9.23854	−0.461351	−0.230675	+	0.973031i	$0.574094\pi$
$402$	0	0
$403$	2.96854	0.147874
$404$	12.7989	0.636769
$405$	0	0
$406$	14.2439	0.706914
$407$	−3.98907	−0.197731
$408$	0	0
$409$	16.0077	0.791532	0.395766	−	0.918351i	$-0.370479\pi$
$409$	16.0077	0.791532	0.395766	+	0.918351i	$0.370479\pi$
$410$	18.9982	0.938256
$411$	0	0
$412$	−8.79561	−0.433328
$413$	9.85978	0.485168
$414$	0	0
$415$	14.2763	0.700797
$416$	−8.27631	−0.405780
$417$	0	0
$418$	0	0
$419$	20.9828	1.02508	0.512538	−	0.858665i	$-0.328706\pi$
$419$	20.9828	1.02508	0.512538	+	0.858665i	$0.328706\pi$
$420$	0	0
$421$	−19.8898	−0.969370	−0.484685	−	0.874689i	$-0.661066\pi$
$421$	−19.8898	−0.969370	−0.484685	+	0.874689i	$0.661066\pi$
$422$	18.5416	0.902590
$423$	0	0
$424$	4.28405	0.208052
$425$	−3.37227	−0.163579
$426$	0	0
$427$	6.65776	0.322191
$428$	−15.7906	−0.763265
$429$	0	0
$430$	25.1411	1.21241
$431$	−37.8949	−1.82533	−0.912666	−	0.408707i	$-0.865980\pi$
$431$	−37.8949	−1.82533	−0.912666	+	0.408707i	$0.865980\pi$
$432$	0	0
$433$	2.01960	0.0970558	0.0485279	−	0.998822i	$-0.484547\pi$
$433$	2.01960	0.0970558	0.0485279	+	0.998822i	$0.484547\pi$
$434$	4.71749	0.226447
$435$	0	0
$436$	15.6783	0.750854
$437$	0	0
$438$	0	0
$439$	−31.3482	−1.49617	−0.748085	−	0.663603i	$-0.769026\pi$
$439$	−31.3482	−1.49617	−0.748085	+	0.663603i	$0.769026\pi$
$440$	2.95636	0.140939
$441$	0	0
$442$	3.08410	0.146696
$443$	17.6486	0.838510	0.419255	−	0.907869i	$-0.362291\pi$
$443$	17.6486	0.838510	0.419255	+	0.907869i	$0.362291\pi$
$444$	0	0
$445$	4.05913	0.192421
$446$	17.3455	0.821335
$447$	0	0
$448$	−13.3746	−0.631892
$449$	−12.2686	−0.578990	−0.289495	−	0.957180i	$-0.593487\pi$
$449$	−12.2686	−0.578990	−0.289495	+	0.957180i	$0.593487\pi$
$450$	0	0
$451$	−3.51073	−0.165314
$452$	13.5030	0.635127
$453$	0	0
$454$	23.2175	1.08965
$455$	−9.85978	−0.462234
$456$	0	0
$457$	19.1875	0.897552	0.448776	−	0.893644i	$-0.351860\pi$
$457$	19.1875	0.897552	0.448776	+	0.893644i	$0.351860\pi$
$458$	−5.16344	−0.241272
$459$	0	0
$460$	−2.73143	−0.127354
$461$	39.4739	1.83848	0.919241	−	0.393694i	$-0.128803\pi$
$461$	39.4739	1.83848	0.919241	+	0.393694i	$0.128803\pi$
$462$	0	0
$463$	14.1857	0.659267	0.329633	−	0.944109i	$-0.393075\pi$
$463$	14.1857	0.659267	0.329633	+	0.944109i	$0.393075\pi$
$464$	0.255777	0.0118742
$465$	0	0
$466$	22.8536	1.05867
$467$	29.7870	1.37838	0.689190	−	0.724581i	$-0.257967\pi$
$467$	29.7870	1.37838	0.689190	+	0.724581i	$0.257967\pi$
$468$	0	0
$469$	12.9436	0.597678
$470$	18.8726	0.870527
$471$	0	0
$472$	10.5466	0.485448
$473$	−4.64590	−0.213619
$474$	0	0
$475$	0	0
$476$	−7.77446	−0.356342
$477$	0	0
$478$	−7.36009	−0.336643
$479$	−0.391874	−0.0179052	−0.00895258	−	0.999960i	$-0.502850\pi$
$479$	−0.391874	−0.0179052	−0.00895258	+	0.999960i	$0.502850\pi$
$480$	0	0
$481$	14.2308	0.648869
$482$	−9.56717	−0.435773
$483$	0	0
$484$	13.2858	0.603900
$485$	0.255777	0.0116143
$486$	0	0
$487$	−24.2327	−1.09809	−0.549043	−	0.835794i	$-0.685008\pi$
$487$	−24.2327	−1.09809	−0.549043	+	0.835794i	$0.685008\pi$
$488$	7.12155	0.322377
$489$	0	0
$490$	−0.0820234	−0.00370544
$491$	34.0779	1.53791	0.768957	−	0.639300i	$-0.220776\pi$
$491$	34.0779	1.53791	0.768957	+	0.639300i	$0.220776\pi$
$492$	0	0
$493$	14.5885	0.657034
$494$	0	0
$495$	0	0
$496$	0.0847118	0.00380367
$497$	−26.7178	−1.19846
$498$	0	0
$499$	−29.5006	−1.32063	−0.660315	−	0.750989i	$-0.729577\pi$
$499$	−29.5006	−1.32063	−0.660315	+	0.750989i	$0.729577\pi$
$500$	−11.1462	−0.498473
$501$	0	0
$502$	−5.45161	−0.243317
$503$	−23.3155	−1.03959	−0.519794	−	0.854292i	$-0.673991\pi$
$503$	−23.3155	−1.03959	−0.519794	+	0.854292i	$0.673991\pi$
$504$	0	0
$505$	26.4192	1.17564
$506$	−0.318201	−0.0141457
$507$	0	0
$508$	6.80396	0.301877
$509$	6.55674	0.290622	0.145311	−	0.989386i	$-0.453582\pi$
$509$	6.55674	0.290622	0.145311	+	0.989386i	$0.453582\pi$
$510$	0	0
$511$	27.1875	1.20270
$512$	−0.473897	−0.0209435
$513$	0	0
$514$	−14.8803	−0.656343
$515$	−18.1557	−0.800036
$516$	0	0
$517$	−3.48751	−0.153381
$518$	22.6150	0.993648
$519$	0	0
$520$	−10.5466	−0.462501
$521$	12.4611	0.545931	0.272965	−	0.962024i	$-0.411996\pi$
$521$	12.4611	0.545931	0.272965	+	0.962024i	$0.411996\pi$
$522$	0	0
$523$	−26.9213	−1.17719	−0.588593	−	0.808430i	$-0.700318\pi$
$523$	−26.9213	−1.17719	−0.588593	+	0.808430i	$0.700318\pi$
$524$	24.0300	1.04976
$525$	0	0
$526$	12.8402	0.559859
$527$	4.83162	0.210469
$528$	0	0
$529$	−22.2267	−0.966377
$530$	3.36184	0.146029
$531$	0	0
$532$	0	0
$533$	12.5243	0.542490
$534$	0	0
$535$	−32.5945	−1.40918
$536$	13.8452	0.598023
$537$	0	0
$538$	15.1753	0.654254
$539$	0.0151573	0.000652872 0
$540$	0	0
$541$	−27.0378	−1.16244	−0.581222	−	0.813745i	$-0.697425\pi$
$541$	−27.0378	−1.16244	−0.581222	+	0.813745i	$0.697425\pi$
$542$	−4.07192	−0.174904
$543$	0	0
$544$	−13.4706	−0.577547
$545$	32.3628	1.38627
$546$	0	0
$547$	−0.443258	−0.0189524	−0.00947618	−	0.999955i	$-0.503016\pi$
$547$	−0.443258	−0.0189524	−0.00947618	+	0.999955i	$0.503016\pi$
$548$	−23.6794	−1.01154
$549$	0	0
$550$	0.510734	0.0217778
$551$	0	0
$552$	0	0
$553$	−43.0265	−1.82967
$554$	−4.15394	−0.176484
$555$	0	0
$556$	−3.89838	−0.165328
$557$	28.4902	1.20717	0.603584	−	0.797299i	$-0.293739\pi$
$557$	28.4902	1.20717	0.603584	+	0.797299i	$0.293739\pi$
$558$	0	0
$559$	16.5740	0.701005
$560$	−0.281364	−0.0118898
$561$	0	0
$562$	−28.7128	−1.21117
$563$	1.98452	0.0836375	0.0418187	−	0.999125i	$-0.486685\pi$
$563$	1.98452	0.0836375	0.0418187	+	0.999125i	$0.486685\pi$
$564$	0	0
$565$	27.8726	1.17261
$566$	−8.60132	−0.361540
$567$	0	0
$568$	−28.5790	−1.19915
$569$	−11.7638	−0.493165	−0.246583	−	0.969122i	$-0.579308\pi$
$569$	−11.7638	−0.493165	−0.246583	+	0.969122i	$0.579308\pi$
$570$	0	0
$571$	3.64227	0.152424	0.0762122	−	0.997092i	$-0.475717\pi$
$571$	3.64227	0.152424	0.0762122	+	0.997092i	$0.475717\pi$
$572$	0.740925	0.0309796
$573$	0	0
$574$	19.9032	0.830744
$575$	−1.24123	−0.0517628
$576$	0	0
$577$	29.3182	1.22053	0.610266	−	0.792196i	$-0.291062\pi$
$577$	29.3182	1.22053	0.610266	+	0.792196i	$0.291062\pi$
$578$	−9.92984	−0.413027
$579$	0	0
$580$	−18.9659	−0.787514
$581$	14.9564	0.620494
$582$	0	0
$583$	−0.621244	−0.0257293
$584$	29.0814	1.20340
$585$	0	0
$586$	3.63640	0.150218
$587$	6.83574	0.282141	0.141071	−	0.990000i	$-0.454946\pi$
$587$	6.83574	0.282141	0.141071	+	0.990000i	$0.454946\pi$
$588$	0	0
$589$	0	0
$590$	8.27631	0.340730
$591$	0	0
$592$	0.406097	0.0166905
$593$	12.1189	0.497662	0.248831	−	0.968547i	$-0.419954\pi$
$593$	12.1189	0.497662	0.248831	+	0.968547i	$0.419954\pi$
$594$	0	0
$595$	−16.0479	−0.657899
$596$	−12.8922	−0.528084
$597$	0	0
$598$	1.13516	0.0464203
$599$	−29.3327	−1.19850	−0.599252	−	0.800561i	$-0.704535\pi$
$599$	−29.3327	−1.19850	−0.599252	+	0.800561i	$0.704535\pi$
$600$	0	0
$601$	−0.820260	−0.0334591	−0.0167295	−	0.999860i	$-0.505325\pi$
$601$	−0.820260	−0.0334591	−0.0167295	+	0.999860i	$0.505325\pi$
$602$	26.3387	1.07349
$603$	0	0
$604$	−11.7716	−0.478978
$605$	27.4243	1.11496
$606$	0	0
$607$	−32.5654	−1.32179	−0.660895	−	0.750479i	$-0.729823\pi$
$607$	−32.5654	−1.32179	−0.660895	+	0.750479i	$0.729823\pi$
$608$	0	0
$609$	0	0
$610$	5.58853	0.226273
$611$	12.4415	0.503329
$612$	0	0
$613$	7.98276	0.322421	0.161210	−	0.986920i	$-0.448460\pi$
$613$	7.98276	0.322421	0.161210	+	0.986920i	$0.448460\pi$
$614$	−5.47565	−0.220979
$615$	0	0
$616$	3.09718	0.124789
$617$	−40.5066	−1.63073	−0.815367	−	0.578944i	$-0.803465\pi$
$617$	−40.5066	−1.63073	−0.815367	+	0.578944i	$0.803465\pi$
$618$	0	0
$619$	−11.4074	−0.458500	−0.229250	−	0.973368i	$-0.573627\pi$
$619$	−11.4074	−0.458500	−0.229250	+	0.973368i	$0.573627\pi$
$620$	−6.28136	−0.252266
$621$	0	0
$622$	−2.27631	−0.0912718
$623$	4.25248	0.170372
$624$	0	0
$625$	−30.0651	−1.20260
$626$	−18.1789	−0.726576
$627$	0	0
$628$	18.8693	0.752966
$629$	23.1622	0.923537
$630$	0	0
$631$	1.07697	0.0428735	0.0214368	−	0.999770i	$-0.493176\pi$
$631$	1.07697	0.0428735	0.0214368	+	0.999770i	$0.493176\pi$
$632$	−46.0238	−1.83073
$633$	0	0
$634$	8.03415	0.319077
$635$	14.0446	0.557342
$636$	0	0
$637$	−0.0540729	−0.00214245
$638$	−2.20945	−0.0874728
$639$	0	0
$640$	17.3259	0.684868
$641$	4.67593	0.184688	0.0923440	−	0.995727i	$-0.470564\pi$
$641$	4.67593	0.184688	0.0923440	+	0.995727i	$0.470564\pi$
$642$	0	0
$643$	−22.6168	−0.891920	−0.445960	−	0.895053i	$-0.647138\pi$
$643$	−22.6168	−0.891920	−0.445960	+	0.895053i	$0.647138\pi$
$644$	−2.86154	−0.112760
$645$	0	0
$646$	0	0
$647$	22.0479	0.866791	0.433396	−	0.901204i	$-0.357315\pi$
$647$	22.0479	0.866791	0.433396	+	0.901204i	$0.357315\pi$
$648$	0	0
$649$	−1.52940	−0.0600342
$650$	−1.82201	−0.0714653
$651$	0	0
$652$	−22.5672	−0.883799
$653$	−38.9796	−1.52539	−0.762694	−	0.646759i	$-0.776124\pi$
$653$	−38.9796	−1.52539	−0.762694	+	0.646759i	$0.776124\pi$
$654$	0	0
$655$	49.6023	1.93812
$656$	0.357401	0.0139542
$657$	0	0
$658$	19.7716	0.770776
$659$	15.7929	0.615205	0.307602	−	0.951515i	$-0.400473\pi$
$659$	15.7929	0.615205	0.307602	+	0.951515i	$0.400473\pi$
$660$	0	0
$661$	−41.0155	−1.59532	−0.797659	−	0.603109i	$-0.793928\pi$
$661$	−41.0155	−1.59532	−0.797659	+	0.603109i	$0.793928\pi$
$662$	14.1530	0.550072
$663$	0	0
$664$	15.9982	0.620852
$665$	0	0
$666$	0	0
$667$	5.36959	0.207911
$668$	28.0847	1.08663
$669$	0	0
$670$	10.8648	0.419745
$671$	−1.03272	−0.0398676
$672$	0	0
$673$	−30.2891	−1.16756	−0.583780	−	0.811912i	$-0.698427\pi$
$673$	−30.2891	−1.16756	−0.583780	+	0.811912i	$0.698427\pi$
$674$	−17.9744	−0.692349
$675$	0	0
$676$	13.3037	0.511679
$677$	2.71419	0.104315	0.0521575	−	0.998639i	$-0.483390\pi$
$677$	2.71419	0.104315	0.0521575	+	0.998639i	$0.483390\pi$
$678$	0	0
$679$	0.267961	0.0102834
$680$	−17.1658	−0.658278
$681$	0	0
$682$	−0.731754	−0.0280203
$683$	32.8803	1.25813	0.629065	−	0.777353i	$-0.283438\pi$
$683$	32.8803	1.25813	0.629065	+	0.777353i	$0.283438\pi$
$684$	0	0
$685$	−48.8786	−1.86755
$686$	16.2433	0.620172
$687$	0	0
$688$	0.472964	0.0180316
$689$	2.21625	0.0844325
$690$	0	0
$691$	−45.4020	−1.72717	−0.863586	−	0.504201i	$-0.831787\pi$
$691$	−45.4020	−1.72717	−0.863586	+	0.504201i	$0.831787\pi$
$692$	−13.2825	−0.504925
$693$	0	0
$694$	−19.4210	−0.737210
$695$	−8.04694	−0.305238
$696$	0	0
$697$	20.3847	0.772127
$698$	−16.2909	−0.616618
$699$	0	0
$700$	4.59297	0.173598
$701$	1.62031	0.0611983	0.0305991	−	0.999532i	$-0.490258\pi$
$701$	1.62031	0.0611983	0.0305991	+	0.999532i	$0.490258\pi$
$702$	0	0
$703$	0	0
$704$	2.07461	0.0781897
$705$	0	0
$706$	27.3566	1.02958
$707$	27.6777	1.04093
$708$	0	0
$709$	36.2449	1.36120	0.680602	−	0.732653i	$-0.261718\pi$
$709$	36.2449	1.36120	0.680602	+	0.732653i	$0.261718\pi$
$710$	−22.4270	−0.841669
$711$	0	0
$712$	4.54872	0.170470
$713$	1.77837	0.0666005
$714$	0	0
$715$	1.52940	0.0571963
$716$	8.04013	0.300474
$717$	0	0
$718$	−14.9504	−0.557943
$719$	6.22256	0.232062	0.116031	−	0.993246i	$-0.462983\pi$
$719$	6.22256	0.232062	0.116031	+	0.993246i	$0.462983\pi$
$720$	0	0
$721$	−19.0205	−0.708362
$722$	0	0
$723$	0	0
$724$	−6.87433	−0.255483
$725$	−8.61856	−0.320085
$726$	0	0
$727$	16.2026	0.600923	0.300461	−	0.953794i	$-0.402859\pi$
$727$	16.2026	0.600923	0.300461	+	0.953794i	$0.402859\pi$
$728$	−11.0490	−0.409504
$729$	0	0
$730$	22.8212	0.844650
$731$	26.9760	0.997742
$732$	0	0
$733$	19.7050	0.727822	0.363911	−	0.931434i	$-0.381441\pi$
$733$	19.7050	0.727822	0.363911	+	0.931434i	$0.381441\pi$
$734$	20.2199	0.746329
$735$	0	0
$736$	−4.95811	−0.182758
$737$	−2.00774	−0.0739561
$738$	0	0
$739$	19.2371	0.707649	0.353824	−	0.935312i	$-0.384881\pi$
$739$	19.2371	0.707649	0.353824	+	0.935312i	$0.384881\pi$
$740$	−30.1121	−1.10694
$741$	0	0
$742$	3.52198	0.129296
$743$	15.0473	0.552031	0.276015	−	0.961153i	$-0.410986\pi$
$743$	15.0473	0.552031	0.276015	+	0.961153i	$0.410986\pi$
$744$	0	0
$745$	−26.6117	−0.974979
$746$	3.01361	0.110336
$747$	0	0
$748$	1.20594	0.0440934
$749$	−34.1471	−1.24771
$750$	0	0
$751$	−27.3236	−0.997051	−0.498526	−	0.866875i	$-0.666125\pi$
$751$	−27.3236	−0.997051	−0.498526	+	0.866875i	$0.666125\pi$
$752$	0.355037	0.0129469
$753$	0	0
$754$	7.88207	0.287048
$755$	−24.2986	−0.884317
$756$	0	0
$757$	−7.32501	−0.266232	−0.133116	−	0.991100i	$-0.542498\pi$
$757$	−7.32501	−0.266232	−0.133116	+	0.991100i	$0.542498\pi$
$758$	−10.7196	−0.389352
$759$	0	0
$760$	0	0
$761$	23.1830	0.840384	0.420192	−	0.907435i	$-0.361963\pi$
$761$	23.1830	0.840384	0.420192	+	0.907435i	$0.361963\pi$
$762$	0	0
$763$	33.9044	1.22742
$764$	−4.00950	−0.145058
$765$	0	0
$766$	−8.14466	−0.294278
$767$	5.45605	0.197007
$768$	0	0
$769$	−4.46791	−0.161117	−0.0805585	−	0.996750i	$-0.525670\pi$
$769$	−4.46791	−0.161117	−0.0805585	+	0.996750i	$0.525670\pi$
$770$	2.43047	0.0875879
$771$	0	0
$772$	1.95481	0.0703553
$773$	−3.97864	−0.143102	−0.0715510	−	0.997437i	$-0.522795\pi$
$773$	−3.97864	−0.143102	−0.0715510	+	0.997437i	$0.522795\pi$
$774$	0	0
$775$	−2.85441	−0.102533
$776$	0.286628	0.0102893
$777$	0	0
$778$	−32.0746	−1.14993
$779$	0	0
$780$	0	0
$781$	4.14433	0.148296
$782$	1.84760	0.0660701
$783$	0	0
$784$	−0.00154305	−5.51090e−5 0
$785$	38.9495	1.39017
$786$	0	0
$787$	−33.3732	−1.18963	−0.594813	−	0.803864i	$-0.702774\pi$
$787$	−33.3732	−1.18963	−0.594813	+	0.803864i	$0.702774\pi$
$788$	−26.0898	−0.929409
$789$	0	0
$790$	−36.1165	−1.28497
$791$	29.2003	1.03824
$792$	0	0
$793$	3.68416	0.130828
$794$	−13.7784	−0.488976
$795$	0	0
$796$	21.8693	0.775136
$797$	3.56036	0.126115	0.0630573	−	0.998010i	$-0.479915\pi$
$797$	3.56036	0.126115	0.0630573	+	0.998010i	$0.479915\pi$
$798$	0	0
$799$	20.2499	0.716390
$800$	7.95811	0.281362
$801$	0	0
$802$	−8.12424	−0.286877
$803$	−4.21719	−0.148821
$804$	0	0
$805$	−5.90673	−0.208185
$806$	2.61049	0.0919506
$807$	0	0
$808$	29.6058	1.04153
$809$	−38.4953	−1.35342	−0.676710	−	0.736249i	$-0.736595\pi$
$809$	−38.4953	−1.35342	−0.676710	+	0.736249i	$0.736595\pi$
$810$	0	0
$811$	−12.9067	−0.453216	−0.226608	−	0.973986i	$-0.572764\pi$
$811$	−12.9067	−0.453216	−0.226608	+	0.973986i	$0.572764\pi$
$812$	−19.8693	−0.697275
$813$	0	0
$814$	−3.50793	−0.122953
$815$	−46.5827	−1.63172
$816$	0	0
$817$	0	0
$818$	14.0770	0.492190
$819$	0	0
$820$	−26.5012	−0.925463
$821$	43.3441	1.51272	0.756360	−	0.654156i	$-0.226976\pi$
$821$	43.3441	1.51272	0.756360	+	0.654156i	$0.226976\pi$
$822$	0	0
$823$	−13.5047	−0.470746	−0.235373	−	0.971905i	$-0.575631\pi$
$823$	−13.5047	−0.470746	−0.235373	+	0.971905i	$0.575631\pi$
$824$	−20.3455	−0.708771
$825$	0	0
$826$	8.67055	0.301687
$827$	−16.6435	−0.578752	−0.289376	−	0.957215i	$-0.593448\pi$
$827$	−16.6435	−0.578752	−0.289376	+	0.957215i	$0.593448\pi$
$828$	0	0
$829$	−48.8667	−1.69721	−0.848605	−	0.529026i	$-0.822557\pi$
$829$	−48.8667	−1.69721	−0.848605	+	0.529026i	$0.822557\pi$
$830$	12.5544	0.435769
$831$	0	0
$832$	−7.40104	−0.256585
$833$	−0.0880095	−0.00304935
$834$	0	0
$835$	57.9718	2.00620
$836$	0	0
$837$	0	0
$838$	18.4519	0.637411
$839$	−36.2121	−1.25018	−0.625091	−	0.780552i	$-0.714938\pi$
$839$	−36.2121	−1.25018	−0.625091	+	0.780552i	$0.714938\pi$
$840$	0	0
$841$	8.28405	0.285657
$842$	−17.4908	−0.602773
$843$	0	0
$844$	−25.8642	−0.890283
$845$	27.4611	0.944691
$846$	0	0
$847$	28.7306	0.987195
$848$	0.0632441	0.00217181
$849$	0	0
$850$	−2.96553	−0.101717
$851$	8.52528	0.292243
$852$	0	0
$853$	50.6408	1.73391	0.866955	−	0.498387i	$-0.166074\pi$
$853$	50.6408	1.73391	0.866955	+	0.498387i	$0.166074\pi$
$854$	5.85473	0.200345
$855$	0	0
$856$	−36.5259	−1.24843
$857$	−44.1611	−1.50851	−0.754257	−	0.656580i	$-0.772003\pi$
$857$	−44.1611	−1.50851	−0.754257	+	0.656580i	$0.772003\pi$
$858$	0	0
$859$	−4.63404	−0.158111	−0.0790556	−	0.996870i	$-0.525190\pi$
$859$	−4.63404	−0.158111	−0.0790556	+	0.996870i	$0.525190\pi$
$860$	−35.0702	−1.19588
$861$	0	0
$862$	−33.3242	−1.13503
$863$	46.2404	1.57404	0.787021	−	0.616926i	$-0.211622\pi$
$863$	46.2404	1.57404	0.787021	+	0.616926i	$0.211622\pi$
$864$	0	0
$865$	−27.4175	−0.932222
$866$	1.77601	0.0603512
$867$	0	0
$868$	−6.58057	−0.223359
$869$	6.67406	0.226402
$870$	0	0
$871$	7.16250	0.242692
$872$	36.2662	1.22813
$873$	0	0
$874$	0	0
$875$	−24.1037	−0.814854
$876$	0	0
$877$	−20.9709	−0.708137	−0.354069	−	0.935219i	$-0.615202\pi$
$877$	−20.9709	−0.708137	−0.354069	+	0.935219i	$0.615202\pi$
$878$	−27.5672	−0.930347
$879$	0	0
$880$	0.0436438	0.00147123
$881$	17.3960	0.586086	0.293043	−	0.956099i	$-0.405332\pi$
$881$	17.3960	0.586086	0.293043	+	0.956099i	$0.405332\pi$
$882$	0	0
$883$	15.7324	0.529436	0.264718	−	0.964326i	$-0.414721\pi$
$883$	15.7324	0.529436	0.264718	+	0.964326i	$0.414721\pi$
$884$	−4.30211	−0.144696
$885$	0	0
$886$	15.5199	0.521401
$887$	23.1114	0.776006	0.388003	−	0.921658i	$-0.373165\pi$
$887$	23.1114	0.776006	0.388003	+	0.921658i	$0.373165\pi$
$888$	0	0
$889$	14.7136	0.493478
$890$	3.56953	0.119651
$891$	0	0
$892$	−24.1958	−0.810136
$893$	0	0
$894$	0	0
$895$	16.5963	0.554752
$896$	18.1513	0.606391
$897$	0	0
$898$	−10.7888	−0.360027
$899$	12.3482	0.411837
$900$	0	0
$901$	3.60719	0.120173
$902$	−3.08729	−0.102795
$903$	0	0
$904$	31.2344	1.03884
$905$	−14.1898	−0.471686
$906$	0	0
$907$	18.5871	0.617174	0.308587	−	0.951196i	$-0.400144\pi$
$907$	18.5871	0.617174	0.308587	+	0.951196i	$0.400144\pi$
$908$	−32.3868	−1.07479
$909$	0	0
$910$	−8.67055	−0.287426
$911$	−4.35410	−0.144258	−0.0721289	−	0.997395i	$-0.522979\pi$
$911$	−4.35410	−0.144258	−0.0721289	+	0.997395i	$0.522979\pi$
$912$	0	0
$913$	−2.31996	−0.0767793
$914$	16.8732	0.558115
$915$	0	0
$916$	7.20264	0.237982
$917$	51.9650	1.71604
$918$	0	0
$919$	−7.82976	−0.258280	−0.129140	−	0.991626i	$-0.541222\pi$
$919$	−7.82976	−0.258280	−0.129140	+	0.991626i	$0.541222\pi$
$920$	−6.31820	−0.208305
$921$	0	0
$922$	34.7128	1.14320
$923$	−14.7847	−0.486644
$924$	0	0
$925$	−13.6837	−0.449916
$926$	12.4747	0.409945
$927$	0	0
$928$	−34.4270	−1.13012
$929$	−2.43882	−0.0800149	−0.0400075	−	0.999199i	$-0.512738\pi$
$929$	−2.43882	−0.0800149	−0.0400075	+	0.999199i	$0.512738\pi$
$930$	0	0
$931$	0	0
$932$	−31.8792	−1.04424
$933$	0	0
$934$	26.1943	0.857103
$935$	2.48927	0.0814077
$936$	0	0
$937$	−45.1046	−1.47350	−0.736752	−	0.676163i	$-0.763642\pi$
$937$	−45.1046	−1.47350	−0.736752	+	0.676163i	$0.763642\pi$
$938$	11.3824	0.371648
$939$	0	0
$940$	−26.3259	−0.858658
$941$	54.4371	1.77460	0.887299	−	0.461195i	$-0.152579\pi$
$941$	54.4371	1.77460	0.887299	+	0.461195i	$0.152579\pi$
$942$	0	0
$943$	7.50299	0.244331
$944$	0.155697	0.00506750
$945$	0	0
$946$	−4.08553	−0.132832
$947$	10.7368	0.348899	0.174450	−	0.984666i	$-0.444185\pi$
$947$	10.7368	0.348899	0.174450	+	0.984666i	$0.444185\pi$
$948$	0	0
$949$	15.0446	0.488368
$950$	0	0
$951$	0	0
$952$	−17.9835	−0.582848
$953$	−9.42190	−0.305205	−0.152603	−	0.988288i	$-0.548765\pi$
$953$	−9.42190	−0.305205	−0.152603	+	0.988288i	$0.548765\pi$
$954$	0	0
$955$	−8.27631	−0.267815
$956$	10.2668	0.332053
$957$	0	0
$958$	−0.344608	−0.0111338
$959$	−51.2069	−1.65356
$960$	0	0
$961$	−26.9103	−0.868076
$962$	12.5144	0.403479
$963$	0	0
$964$	13.3455	0.429831
$965$	4.03508	0.129894
$966$	0	0
$967$	27.4029	0.881218	0.440609	−	0.897699i	$-0.354762\pi$
$967$	27.4029	0.881218	0.440609	+	0.897699i	$0.354762\pi$
$968$	30.7320	0.987765
$969$	0	0
$970$	0.224927	0.00722197
$971$	−20.0942	−0.644854	−0.322427	−	0.946594i	$-0.604499\pi$
$971$	−20.0942	−0.644854	−0.322427	+	0.946594i	$0.604499\pi$
$972$	0	0
$973$	−8.43025	−0.270262
$974$	−21.3099	−0.682812
$975$	0	0
$976$	0.105133	0.00336523
$977$	−14.1274	−0.451976	−0.225988	−	0.974130i	$-0.572561\pi$
$977$	−14.1274	−0.451976	−0.225988	+	0.974130i	$0.572561\pi$
$978$	0	0
$979$	−0.659623	−0.0210817
$980$	0.114417	0.00365492
$981$	0	0
$982$	29.9676	0.956305
$983$	33.2550	1.06067	0.530334	−	0.847789i	$-0.322066\pi$
$983$	33.2550	1.06067	0.530334	+	0.847789i	$0.322066\pi$
$984$	0	0
$985$	−53.8539	−1.71593
$986$	12.8289	0.408557
$987$	0	0
$988$	0	0
$989$	9.92902	0.315724
$990$	0	0
$991$	−27.0642	−0.859722	−0.429861	−	0.902895i	$-0.641437\pi$
$991$	−27.0642	−0.859722	−0.429861	+	0.902895i	$0.641437\pi$
$992$	−11.4020	−0.362013
$993$	0	0
$994$	−23.4953	−0.745224
$995$	45.1421	1.43110
$996$	0	0
$997$	−54.5049	−1.72619	−0.863093	−	0.505045i	$-0.831476\pi$
$997$	−54.5049	−1.72619	−0.863093	+	0.505045i	$0.831476\pi$
$998$	−25.9424	−0.821193
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9747.2.a.u.1.3		3
3.2	odd	2		9747.2.a.bd.1.1		3
19.6	even	9		513.2.y.c.55.1	yes	6
19.16	even	9		513.2.y.c.28.1	yes	6
19.18	odd	2		9747.2.a.bb.1.1		3
57.35	odd	18		513.2.y.a.28.1	✓	6
57.44	odd	18		513.2.y.a.55.1	yes	6
57.56	even	2		9747.2.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.y.a.28.1	✓	6	57.35	odd	18
513.2.y.a.55.1	yes	6	57.44	odd	18
513.2.y.c.28.1	yes	6	19.16	even	9
513.2.y.c.55.1	yes	6	19.6	even	9
9747.2.a.u.1.3		3	1.1	even	1	trivial
9747.2.a.v.1.3		3	57.56	even	2
9747.2.a.bb.1.1		3	19.18	odd	2
9747.2.a.bd.1.1		3	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9747 = 3^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9747.a (trivial)

\(f(q)\)	\(=\)	\(q+0.879385 q^{2} -1.22668 q^{4} -2.53209 q^{5} -2.65270 q^{7} -2.83750 q^{8} -2.22668 q^{10} +0.411474 q^{11} -1.46791 q^{13} -2.33275 q^{14} -0.0418891 q^{16} -2.38919 q^{17} +3.10607 q^{20} +0.361844 q^{22} -0.879385 q^{23} +1.41147 q^{25} -1.29086 q^{26} +3.25402 q^{28} -6.10607 q^{29} -2.02229 q^{31} +5.63816 q^{32} -2.10101 q^{34} +6.71688 q^{35} -9.69459 q^{37} +7.18479 q^{40} -8.53209 q^{41} -11.2909 q^{43} -0.504748 q^{44} -0.773318 q^{46} -8.47565 q^{47} +0.0368366 q^{49} +1.24123 q^{50} +1.80066 q^{52} -1.50980 q^{53} -1.04189 q^{55} +7.52704 q^{56} -5.36959 q^{58} -3.71688 q^{59} -2.50980 q^{61} -1.77837 q^{62} +5.04189 q^{64} +3.71688 q^{65} -4.87939 q^{67} +2.93077 q^{68} +5.90673 q^{70} +10.0719 q^{71} -10.2490 q^{73} -8.52528 q^{74} -1.09152 q^{77} +16.2199 q^{79} +0.106067 q^{80} -7.50299 q^{82} -5.63816 q^{83} +6.04963 q^{85} -9.92902 q^{86} -1.16756 q^{88} -1.60307 q^{89} +3.89393 q^{91} +1.07873 q^{92} -7.45336 q^{94} -0.101014 q^{97} +0.0323936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 9 q^{7} - 6 q^{8} - 9 q^{11} - 9 q^{13} + 12 q^{14} + 3 q^{16} - 3 q^{17} - 3 q^{20} + 18 q^{22} + 3 q^{23} - 6 q^{25} + 12 q^{26} - 18 q^{28} - 6 q^{29} - 9 q^{34} + 12 q^{35}+ \cdots - 36 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 9 * q^7 - 6 * q^8 - 9 * q^11 - 9 * q^13 + 12 * q^14 + 3 * q^16 - 3 * q^17 - 3 * q^20 + 18 * q^22 + 3 * q^23 - 6 * q^25 + 12 * q^26 - 18 * q^28 - 6 * q^29 - 9 * q^34 + 12 * q^35 - 27 * q^37 + 18 * q^40 - 21 * q^41 - 18 * q^43 - 18 * q^44 - 9 * q^46 - 6 * q^47 + 12 * q^49 + 15 * q^50 - 9 * q^52 - 6 * q^53 + 33 * q^56 - 9 * q^58 - 3 * q^59 - 9 * q^61 + 3 * q^62 + 12 * q^64 + 3 * q^65 - 9 * q^67 + 33 * q^68 - 9 * q^70 - 3 * q^71 - 18 * q^73 + 21 * q^74 + 27 * q^77 - 12 * q^80 + 18 * q^82 - 9 * q^85 + 3 * q^86 + 9 * q^88 - 33 * q^89 + 24 * q^91 + 12 * q^92 - 9 * q^94 - 3 * q^97 - 36 * q^98

Introduction

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