LMFDB - Embedded newform 8649.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68149 q^{2} +5.19036 q^{4} -1.12713 q^{5} +2.95026 q^{7} -8.55491 q^{8} +O(q^{10})$	$q-2.68149 q^{2} +5.19036 q^{4} -1.12713 q^{5} +2.95026 q^{7} -8.55491 q^{8} +3.02238 q^{10} +1.95024 q^{11} +3.92532 q^{13} -7.91108 q^{14} +12.5591 q^{16} +5.68498 q^{17} +7.53246 q^{19} -5.85021 q^{20} -5.22953 q^{22} -3.03484 q^{23} -3.72958 q^{25} -10.5257 q^{26} +15.3129 q^{28} -10.0921 q^{29} -16.5673 q^{32} -15.2442 q^{34} -3.32532 q^{35} -4.71938 q^{37} -20.1982 q^{38} +9.64249 q^{40} +6.43404 q^{41} -5.07118 q^{43} +10.1224 q^{44} +8.13789 q^{46} +4.47567 q^{47} +1.70403 q^{49} +10.0008 q^{50} +20.3738 q^{52} +12.6084 q^{53} -2.19817 q^{55} -25.2392 q^{56} +27.0618 q^{58} +10.7595 q^{59} +0.00914068 q^{61} +19.3068 q^{64} -4.42434 q^{65} -4.38073 q^{67} +29.5071 q^{68} +8.91680 q^{70} +4.17978 q^{71} +9.37006 q^{73} +12.6550 q^{74} +39.0962 q^{76} +5.75370 q^{77} +12.5173 q^{79} -14.1558 q^{80} -17.2528 q^{82} +2.60577 q^{83} -6.40770 q^{85} +13.5983 q^{86} -16.6841 q^{88} -2.14184 q^{89} +11.5807 q^{91} -15.7519 q^{92} -12.0014 q^{94} -8.49005 q^{95} +6.11825 q^{97} -4.56934 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7}+O(q^{10}) $ 24 * q + 32 * q^4 + 16 * q^7	$ 24 q + 32 q^{4} + 16 q^{7} + 40 q^{10} + 64 q^{16} + 32 q^{19} + 24 q^{25} + 104 q^{28} + 160 q^{40} - 40 q^{49} + 136 q^{64} + 80 q^{67} + 72 q^{70} + 120 q^{76} - 152 q^{82} + 80 q^{94} + 32 q^{97}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 40 * q^10 + 64 * q^16 + 32 * q^19 + 24 * q^25 + 104 * q^28 + 160 * q^40 - 40 * q^49 + 136 * q^64 + 80 * q^67 + 72 * q^70 + 120 * q^76 - 152 * q^82 + 80 * q^94 + 32 * 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 $

$2$	−2.68149	−1.89610	−0.948048	−	0.318127i	$-0.896946\pi$
$2$	−2.68149	−1.89610	−0.948048	+	0.318127i	$0.896946\pi$
$3$	0	0
$3$	0	0
$4$	5.19036	2.59518
$5$	−1.12713	−0.504067	−0.252034	−	0.967718i	$-0.581099\pi$
$5$	−1.12713	−0.504067	−0.252034	+	0.967718i	$0.581099\pi$
$6$	0	0
$7$	2.95026	1.11509	0.557547	−	0.830146i	$-0.311743\pi$
$7$	2.95026	1.11509	0.557547	+	0.830146i	$0.311743\pi$
$8$	−8.55491	−3.02462
$9$	0	0
$10$	3.02238	0.955760
$11$	1.95024	0.588019	0.294009	−	0.955803i	$-0.405010\pi$
$11$	1.95024	0.588019	0.294009	+	0.955803i	$0.405010\pi$
$12$	0	0
$13$	3.92532	1.08869	0.544344	−	0.838862i	$-0.316779\pi$
$13$	3.92532	1.08869	0.544344	+	0.838862i	$0.316779\pi$
$14$	−7.91108	−2.11432
$15$	0	0
$16$	12.5591	3.13979
$17$	5.68498	1.37881	0.689405	−	0.724376i	$-0.257872\pi$
$17$	5.68498	1.37881	0.689405	+	0.724376i	$0.257872\pi$
$18$	0	0
$19$	7.53246	1.72806	0.864032	−	0.503436i	$-0.167931\pi$
$19$	7.53246	1.72806	0.864032	+	0.503436i	$0.167931\pi$
$20$	−5.85021	−1.30815
$21$	0	0
$22$	−5.22953	−1.11494
$23$	−3.03484	−0.632809	−0.316404	−	0.948624i	$-0.602476\pi$
$23$	−3.03484	−0.632809	−0.316404	+	0.948624i	$0.602476\pi$
$24$	0	0
$25$	−3.72958	−0.745916
$26$	−10.5257	−2.06426
$27$	0	0
$28$	15.3129	2.89387
$29$	−10.0921	−1.87406	−0.937028	−	0.349255i	$-0.886435\pi$
$29$	−10.0921	−1.87406	−0.937028	+	0.349255i	$0.886435\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−16.5673	−2.92872
$33$	0	0
$34$	−15.2442	−2.61436
$35$	−3.32532	−0.562082
$36$	0	0
$37$	−4.71938	−0.775862	−0.387931	−	0.921688i	$-0.626810\pi$
$37$	−4.71938	−0.775862	−0.387931	+	0.921688i	$0.626810\pi$
$38$	−20.1982	−3.27658
$39$	0	0
$40$	9.64249	1.52461
$41$	6.43404	1.00483	0.502414	−	0.864627i	$-0.332445\pi$
$41$	6.43404	1.00483	0.502414	+	0.864627i	$0.332445\pi$
$42$	0	0
$43$	−5.07118	−0.773348	−0.386674	−	0.922217i	$-0.626376\pi$
$43$	−5.07118	−0.773348	−0.386674	+	0.922217i	$0.626376\pi$
$44$	10.1224	1.52602
$45$	0	0
$46$	8.13789	1.19987
$47$	4.47567	0.652844	0.326422	−	0.945224i	$-0.394157\pi$
$47$	4.47567	0.652844	0.326422	+	0.945224i	$0.394157\pi$
$48$	0	0
$49$	1.70403	0.243433
$50$	10.0008	1.41433
$51$	0	0
$52$	20.3738	2.82534
$53$	12.6084	1.73189	0.865945	−	0.500139i	$-0.166718\pi$
$53$	12.6084	1.73189	0.865945	+	0.500139i	$0.166718\pi$
$54$	0	0
$55$	−2.19817	−0.296401
$56$	−25.2392	−3.37273
$57$	0	0
$58$	27.0618	3.55339
$59$	10.7595	1.40077	0.700384	−	0.713767i	$-0.253012\pi$
$59$	10.7595	1.40077	0.700384	+	0.713767i	$0.253012\pi$
$60$	0	0
$61$	0.00914068	0.00117034	0.000585172	−	1.00000i	$-0.499814\pi$
$61$	0.00914068	0.00117034	0.000585172		1.00000i	$0.499814\pi$
$62$	0	0
$63$	0	0
$64$	19.3068	2.41335
$65$	−4.42434	−0.548772
$66$	0	0
$67$	−4.38073	−0.535191	−0.267596	−	0.963531i	$-0.586229\pi$
$67$	−4.38073	−0.535191	−0.267596	+	0.963531i	$0.586229\pi$
$68$	29.5071	3.57826
$69$	0	0
$70$	8.91680	1.06576
$71$	4.17978	0.496048	0.248024	−	0.968754i	$-0.420219\pi$
$71$	4.17978	0.496048	0.248024	+	0.968754i	$0.420219\pi$
$72$	0	0
$73$	9.37006	1.09668	0.548341	−	0.836255i	$-0.315259\pi$
$73$	9.37006	1.09668	0.548341	+	0.836255i	$0.315259\pi$
$74$	12.6550	1.47111
$75$	0	0
$76$	39.0962	4.48464
$77$	5.75370	0.655696
$78$	0	0
$79$	12.5173	1.40831	0.704156	−	0.710046i	$-0.251326\pi$
$79$	12.5173	1.40831	0.704156	+	0.710046i	$0.251326\pi$
$80$	−14.1558	−1.58266
$81$	0	0
$82$	−17.2528	−1.90525
$83$	2.60577	0.286021	0.143010	−	0.989721i	$-0.454322\pi$
$83$	2.60577	0.286021	0.143010	+	0.989721i	$0.454322\pi$
$84$	0	0
$85$	−6.40770	−0.695013
$86$	13.5983	1.46634
$87$	0	0
$88$	−16.6841	−1.77853
$89$	−2.14184	−0.227034	−0.113517	−	0.993536i	$-0.536212\pi$
$89$	−2.14184	−0.227034	−0.113517	+	0.993536i	$0.536212\pi$
$90$	0	0
$91$	11.5807	1.21399
$92$	−15.7519	−1.64225
$93$	0	0
$94$	−12.0014	−1.23785
$95$	−8.49005	−0.871061
$96$	0	0
$97$	6.11825	0.621214	0.310607	−	0.950538i	$-0.399468\pi$
$97$	6.11825	0.621214	0.310607	+	0.950538i	$0.399468\pi$
$98$	−4.56934	−0.461573
$99$	0	0
$100$	−19.3579	−1.93579
$101$	−3.61530	−0.359736	−0.179868	−	0.983691i	$-0.557567\pi$
$101$	−3.61530	−0.359736	−0.179868	+	0.983691i	$0.557567\pi$
$102$	0	0
$103$	−9.91485	−0.976939	−0.488470	−	0.872581i	$-0.662445\pi$
$103$	−9.91485	−0.976939	−0.488470	+	0.872581i	$0.662445\pi$
$104$	−33.5808	−3.29287
$105$	0	0
$106$	−33.8091	−3.28383
$107$	−14.4589	−1.39780	−0.698899	−	0.715220i	$-0.746327\pi$
$107$	−14.4589	−1.39780	−0.698899	+	0.715220i	$0.746327\pi$
$108$	0	0
$109$	6.25027	0.598667	0.299334	−	0.954148i	$-0.403236\pi$
$109$	6.25027	0.598667	0.299334	+	0.954148i	$0.403236\pi$
$110$	5.89435	0.562005
$111$	0	0
$112$	37.0527	3.50116
$113$	5.50145	0.517533	0.258766	−	0.965940i	$-0.416684\pi$
$113$	5.50145	0.517533	0.258766	+	0.965940i	$0.416684\pi$
$114$	0	0
$115$	3.42066	0.318978
$116$	−52.3817	−4.86351
$117$	0	0
$118$	−28.8514	−2.65599
$119$	16.7722	1.53750
$120$	0	0
$121$	−7.19658	−0.654234
$122$	−0.0245106	−0.00221909
$123$	0	0
$124$	0	0
$125$	9.83936	0.880059
$126$	0	0
$127$	7.06152	0.626609	0.313304	−	0.949653i	$-0.398564\pi$
$127$	7.06152	0.626609	0.313304	+	0.949653i	$0.398564\pi$
$128$	−18.6362	−1.64722
$129$	0	0
$130$	11.8638	1.04052
$131$	9.69758	0.847281	0.423641	−	0.905830i	$-0.360752\pi$
$131$	9.69758	0.847281	0.423641	+	0.905830i	$0.360752\pi$
$132$	0	0
$133$	22.2227	1.92695
$134$	11.7469	1.01477
$135$	0	0
$136$	−48.6345	−4.17038
$137$	5.62594	0.480656	0.240328	−	0.970692i	$-0.422745\pi$
$137$	5.62594	0.480656	0.240328	+	0.970692i	$0.422745\pi$
$138$	0	0
$139$	−10.6204	−0.900813	−0.450407	−	0.892823i	$-0.648721\pi$
$139$	−10.6204	−0.900813	−0.450407	+	0.892823i	$0.648721\pi$
$140$	−17.2596	−1.45870
$141$	0	0
$142$	−11.2080	−0.940556
$143$	7.65531	0.640169
$144$	0	0
$145$	11.3751	0.944650
$146$	−25.1257	−2.07942
$147$	0	0
$148$	−24.4953	−2.01350
$149$	−6.78436	−0.555796	−0.277898	−	0.960611i	$-0.589638\pi$
$149$	−6.78436	−0.555796	−0.277898	+	0.960611i	$0.589638\pi$
$150$	0	0
$151$	−1.56246	−0.127151	−0.0635754	−	0.997977i	$-0.520250\pi$
$151$	−1.56246	−0.127151	−0.0635754	+	0.997977i	$0.520250\pi$
$152$	−64.4395	−5.22674
$153$	0	0
$154$	−15.4285	−1.24326
$155$	0	0
$156$	0	0
$157$	20.0168	1.59752	0.798759	−	0.601651i	$-0.205490\pi$
$157$	20.0168	1.59752	0.798759	+	0.601651i	$0.205490\pi$
$158$	−33.5651	−2.67029
$159$	0	0
$160$	18.6735	1.47627
$161$	−8.95358	−0.705641
$162$	0	0
$163$	1.34686	0.105494	0.0527471	−	0.998608i	$-0.483202\pi$
$163$	1.34686	0.105494	0.0527471	+	0.998608i	$0.483202\pi$
$164$	33.3950	2.60771
$165$	0	0
$166$	−6.98734	−0.542323
$167$	6.44871	0.499016	0.249508	−	0.968373i	$-0.419731\pi$
$167$	6.44871	0.499016	0.249508	+	0.968373i	$0.419731\pi$
$168$	0	0
$169$	2.40815	0.185243
$170$	17.1822	1.31781
$171$	0	0
$172$	−26.3213	−2.00698
$173$	−11.4768	−0.872563	−0.436282	−	0.899810i	$-0.643705\pi$
$173$	−11.4768	−0.872563	−0.436282	+	0.899810i	$0.643705\pi$
$174$	0	0
$175$	−11.0032	−0.831766
$176$	24.4933	1.84625
$177$	0	0
$178$	5.74330	0.430479
$179$	15.1869	1.13512	0.567562	−	0.823330i	$-0.307887\pi$
$179$	15.1869	1.13512	0.567562	+	0.823330i	$0.307887\pi$
$180$	0	0
$181$	6.07082	0.451241	0.225620	−	0.974215i	$-0.427559\pi$
$181$	6.07082	0.451241	0.225620	+	0.974215i	$0.427559\pi$
$182$	−31.0535	−2.30184
$183$	0	0
$184$	25.9628	1.91400
$185$	5.31935	0.391087
$186$	0	0
$187$	11.0871	0.810766
$188$	23.2304	1.69425
$189$	0	0
$190$	22.7659	1.65161
$191$	1.21988	0.0882671	0.0441336	−	0.999026i	$-0.485947\pi$
$191$	1.21988	0.0882671	0.0441336	+	0.999026i	$0.485947\pi$
$192$	0	0
$193$	3.58709	0.258205	0.129102	−	0.991631i	$-0.458790\pi$
$193$	3.58709	0.258205	0.129102	+	0.991631i	$0.458790\pi$
$194$	−16.4060	−1.17788
$195$	0	0
$196$	8.84454	0.631753
$197$	−25.3434	−1.80564	−0.902822	−	0.430013i	$-0.858509\pi$
$197$	−25.3434	−1.80564	−0.902822	+	0.430013i	$0.858509\pi$
$198$	0	0
$199$	10.5214	0.745840	0.372920	−	0.927863i	$-0.378357\pi$
$199$	10.5214	0.745840	0.372920	+	0.927863i	$0.378357\pi$
$200$	31.9063	2.25611
$201$	0	0
$202$	9.69437	0.682094
$203$	−29.7743	−2.08975
$204$	0	0
$205$	−7.25198	−0.506500
$206$	26.5865	1.85237
$207$	0	0
$208$	49.2987	3.41825
$209$	14.6901	1.01613
$210$	0	0
$211$	6.99403	0.481489	0.240744	−	0.970589i	$-0.422608\pi$
$211$	6.99403	0.481489	0.240744	+	0.970589i	$0.422608\pi$
$212$	65.4420	4.49457
$213$	0	0
$214$	38.7714	2.65036
$215$	5.71587	0.389819
$216$	0	0
$217$	0	0
$218$	−16.7600	−1.13513
$219$	0	0
$220$	−11.4093	−0.769214
$221$	22.3154	1.50109
$222$	0	0
$223$	−21.5961	−1.44618	−0.723091	−	0.690753i	$-0.757279\pi$
$223$	−21.5961	−1.44618	−0.723091	+	0.690753i	$0.757279\pi$
$224$	−48.8780	−3.26580
$225$	0	0
$226$	−14.7520	−0.981292
$227$	−4.68853	−0.311189	−0.155594	−	0.987821i	$-0.549729\pi$
$227$	−4.68853	−0.311189	−0.155594	+	0.987821i	$0.549729\pi$
$228$	0	0
$229$	7.49204	0.495088	0.247544	−	0.968877i	$-0.420377\pi$
$229$	7.49204	0.495088	0.247544	+	0.968877i	$0.420377\pi$
$230$	−9.17244	−0.604813
$231$	0	0
$232$	86.3370	5.66830
$233$	22.5784	1.47916	0.739581	−	0.673067i	$-0.235024\pi$
$233$	22.5784	1.47916	0.739581	+	0.673067i	$0.235024\pi$
$234$	0	0
$235$	−5.04466	−0.329077
$236$	55.8457	3.63525
$237$	0	0
$238$	−44.9743	−2.91525
$239$	−20.0319	−1.29576	−0.647878	−	0.761744i	$-0.724344\pi$
$239$	−20.0319	−1.29576	−0.647878	+	0.761744i	$0.724344\pi$
$240$	0	0
$241$	14.3428	0.923898	0.461949	−	0.886906i	$-0.347150\pi$
$241$	14.3428	0.923898	0.461949	+	0.886906i	$0.347150\pi$
$242$	19.2975	1.24049
$243$	0	0
$244$	0.0474435	0.00303726
$245$	−1.92066	−0.122707
$246$	0	0
$247$	29.5673	1.88132
$248$	0	0
$249$	0	0
$250$	−26.3841	−1.66868
$251$	−22.4401	−1.41640	−0.708202	−	0.706010i	$-0.750493\pi$
$251$	−22.4401	−1.41640	−0.708202	+	0.706010i	$0.750493\pi$
$252$	0	0
$253$	−5.91866	−0.372103
$254$	−18.9354	−1.18811
$255$	0	0
$256$	11.3591	0.709944
$257$	−14.3954	−0.897962	−0.448981	−	0.893541i	$-0.648213\pi$
$257$	−14.3954	−0.897962	−0.448981	+	0.893541i	$0.648213\pi$
$258$	0	0
$259$	−13.9234	−0.865159
$260$	−22.9639	−1.42416
$261$	0	0
$262$	−26.0039	−1.60653
$263$	28.0889	1.73204	0.866018	−	0.500012i	$-0.166671\pi$
$263$	28.0889	1.73204	0.866018	+	0.500012i	$0.166671\pi$
$264$	0	0
$265$	−14.2112	−0.872989
$266$	−59.5899	−3.65369
$267$	0	0
$268$	−22.7376	−1.38892
$269$	−4.90517	−0.299073	−0.149537	−	0.988756i	$-0.547778\pi$
$269$	−4.90517	−0.299073	−0.149537	+	0.988756i	$0.547778\pi$
$270$	0	0
$271$	−4.77833	−0.290263	−0.145131	−	0.989412i	$-0.546360\pi$
$271$	−4.77833	−0.290263	−0.145131	+	0.989412i	$0.546360\pi$
$272$	71.3985	4.32917
$273$	0	0
$274$	−15.0859	−0.911371
$275$	−7.27357	−0.438613
$276$	0	0
$277$	−25.9800	−1.56099	−0.780493	−	0.625164i	$-0.785032\pi$
$277$	−25.9800	−1.56099	−0.780493	+	0.625164i	$0.785032\pi$
$278$	28.4785	1.70803
$279$	0	0
$280$	28.4478	1.70008
$281$	2.82248	0.168375	0.0841876	−	0.996450i	$-0.473171\pi$
$281$	2.82248	0.168375	0.0841876	+	0.996450i	$0.473171\pi$
$282$	0	0
$283$	−23.0021	−1.36733	−0.683667	−	0.729794i	$-0.739616\pi$
$283$	−23.0021	−1.36733	−0.683667	+	0.729794i	$0.739616\pi$
$284$	21.6946	1.28734
$285$	0	0
$286$	−20.5276	−1.21382
$287$	18.9821	1.12048
$288$	0	0
$289$	15.3190	0.901118
$290$	−30.5021	−1.79115
$291$	0	0
$292$	48.6340	2.84609
$293$	−6.29912	−0.367999	−0.183999	−	0.982926i	$-0.558904\pi$
$293$	−6.29912	−0.367999	−0.183999	+	0.982926i	$0.558904\pi$
$294$	0	0
$295$	−12.1273	−0.706081
$296$	40.3739	2.34669
$297$	0	0
$298$	18.1922	1.05384
$299$	−11.9127	−0.688931
$300$	0	0
$301$	−14.9613	−0.862355
$302$	4.18970	0.241090
$303$	0	0
$304$	94.6013	5.42575
$305$	−0.0103027	−0.000589932 0
$306$	0	0
$307$	−4.37258	−0.249556	−0.124778	−	0.992185i	$-0.539822\pi$
$307$	−4.37258	−0.249556	−0.124778	+	0.992185i	$0.539822\pi$
$308$	29.8638	1.70165
$309$	0	0
$310$	0	0
$311$	−0.612118	−0.0347100	−0.0173550	−	0.999849i	$-0.505525\pi$
$311$	−0.612118	−0.0347100	−0.0173550	+	0.999849i	$0.505525\pi$
$312$	0	0
$313$	−31.5070	−1.78088	−0.890442	−	0.455097i	$-0.849604\pi$
$313$	−31.5070	−1.78088	−0.890442	+	0.455097i	$0.849604\pi$
$314$	−53.6749	−3.02905
$315$	0	0
$316$	64.9696	3.65482
$317$	10.7255	0.602407	0.301203	−	0.953560i	$-0.402612\pi$
$317$	10.7255	0.602407	0.301203	+	0.953560i	$0.402612\pi$
$318$	0	0
$319$	−19.6820	−1.10198
$320$	−21.7612	−1.21649
$321$	0	0
$322$	24.0089	1.33796
$323$	42.8219	2.38267
$324$	0	0
$325$	−14.6398	−0.812071
$326$	−3.61158	−0.200027
$327$	0	0
$328$	−55.0426	−3.03922
$329$	13.2044	0.727982
$330$	0	0
$331$	−3.55672	−0.195495	−0.0977476	−	0.995211i	$-0.531164\pi$
$331$	−3.55672	−0.195495	−0.0977476	+	0.995211i	$0.531164\pi$
$332$	13.5249	0.742276
$333$	0	0
$334$	−17.2921	−0.946183
$335$	4.93764	0.269772
$336$	0	0
$337$	−10.7961	−0.588099	−0.294050	−	0.955790i	$-0.595003\pi$
$337$	−10.7961	−0.588099	−0.294050	+	0.955790i	$0.595003\pi$
$338$	−6.45743	−0.351238
$339$	0	0
$340$	−33.2583	−1.80369
$341$	0	0
$342$	0	0
$343$	−15.6245	−0.843643
$344$	43.3835	2.33908
$345$	0	0
$346$	30.7748	1.65446
$347$	−8.32989	−0.447172	−0.223586	−	0.974684i	$-0.571776\pi$
$347$	−8.32989	−0.447172	−0.223586	+	0.974684i	$0.571776\pi$
$348$	0	0
$349$	23.0335	1.23295	0.616477	−	0.787373i	$-0.288559\pi$
$349$	23.0335	1.23295	0.616477	+	0.787373i	$0.288559\pi$
$350$	29.5050	1.57711
$351$	0	0
$352$	−32.3102	−1.72214
$353$	8.80770	0.468787	0.234393	−	0.972142i	$-0.424690\pi$
$353$	8.80770	0.468787	0.234393	+	0.972142i	$0.424690\pi$
$354$	0	0
$355$	−4.71115	−0.250042
$356$	−11.1169	−0.589195
$357$	0	0
$358$	−40.7235	−2.15231
$359$	32.5477	1.71780	0.858901	−	0.512141i	$-0.171148\pi$
$359$	32.5477	1.71780	0.858901	+	0.512141i	$0.171148\pi$
$360$	0	0
$361$	37.7379	1.98621
$362$	−16.2788	−0.855596
$363$	0	0
$364$	60.1081	3.15052
$365$	−10.5613	−0.552802
$366$	0	0
$367$	−6.76280	−0.353015	−0.176508	−	0.984299i	$-0.556480\pi$
$367$	−6.76280	−0.353015	−0.176508	+	0.984299i	$0.556480\pi$
$368$	−38.1151	−1.98688
$369$	0	0
$370$	−14.2638	−0.741538
$371$	37.1979	1.93122
$372$	0	0
$373$	13.9923	0.724493	0.362246	−	0.932082i	$-0.382010\pi$
$373$	13.9923	0.724493	0.362246	+	0.932082i	$0.382010\pi$
$374$	−29.7298	−1.53729
$375$	0	0
$376$	−38.2890	−1.97460
$377$	−39.6147	−2.04026
$378$	0	0
$379$	7.90073	0.405833	0.202917	−	0.979196i	$-0.434958\pi$
$379$	7.90073	0.405833	0.202917	+	0.979196i	$0.434958\pi$
$380$	−44.0664	−2.26056
$381$	0	0
$382$	−3.27108	−0.167363
$383$	−16.1566	−0.825563	−0.412781	−	0.910830i	$-0.635443\pi$
$383$	−16.1566	−0.825563	−0.412781	+	0.910830i	$0.635443\pi$
$384$	0	0
$385$	−6.48516	−0.330515
$386$	−9.61874	−0.489581
$387$	0	0
$388$	31.7559	1.61216
$389$	18.3624	0.931010	0.465505	−	0.885045i	$-0.345873\pi$
$389$	18.3624	0.931010	0.465505	+	0.885045i	$0.345873\pi$
$390$	0	0
$391$	−17.2530	−0.872523
$392$	−14.5778	−0.736292
$393$	0	0
$394$	67.9581	3.42368
$395$	−14.1087	−0.709883
$396$	0	0
$397$	22.4811	1.12829	0.564146	−	0.825675i	$-0.309206\pi$
$397$	22.4811	1.12829	0.564146	+	0.825675i	$0.309206\pi$
$398$	−28.2129	−1.41419
$399$	0	0
$400$	−46.8404	−2.34202
$401$	−10.0818	−0.503461	−0.251730	−	0.967797i	$-0.581000\pi$
$401$	−10.0818	−0.503461	−0.251730	+	0.967797i	$0.581000\pi$
$402$	0	0
$403$	0	0
$404$	−18.7647	−0.933580
$405$	0	0
$406$	79.8394	3.96236
$407$	−9.20392	−0.456221
$408$	0	0
$409$	3.06856	0.151731	0.0758653	−	0.997118i	$-0.475828\pi$
$409$	3.06856	0.151731	0.0758653	+	0.997118i	$0.475828\pi$
$410$	19.4461	0.960374
$411$	0	0
$412$	−51.4617	−2.53534
$413$	31.7433	1.56199
$414$	0	0
$415$	−2.93704	−0.144174
$416$	−65.0322	−3.18846
$417$	0	0
$418$	−39.3912	−1.92669
$419$	−19.5258	−0.953897	−0.476948	−	0.878931i	$-0.658257\pi$
$419$	−19.5258	−0.953897	−0.476948	+	0.878931i	$0.658257\pi$
$420$	0	0
$421$	4.77236	0.232591	0.116295	−	0.993215i	$-0.462898\pi$
$421$	4.77236	0.232591	0.116295	+	0.993215i	$0.462898\pi$
$422$	−18.7544	−0.912949
$423$	0	0
$424$	−107.863	−5.23831
$425$	−21.2026	−1.02848
$426$	0	0
$427$	0.0269674	0.00130504
$428$	−75.0472	−3.62754
$429$	0	0
$430$	−15.3270	−0.739135
$431$	19.9949	0.963121	0.481560	−	0.876413i	$-0.340070\pi$
$431$	19.9949	0.963121	0.481560	+	0.876413i	$0.340070\pi$
$432$	0	0
$433$	−31.9113	−1.53356	−0.766781	−	0.641909i	$-0.778143\pi$
$433$	−31.9113	−1.53356	−0.766781	+	0.641909i	$0.778143\pi$
$434$	0	0
$435$	0	0
$436$	32.4412	1.55365
$437$	−22.8598	−1.09353
$438$	0	0
$439$	4.73821	0.226142	0.113071	−	0.993587i	$-0.463931\pi$
$439$	4.73821	0.226142	0.113071	+	0.993587i	$0.463931\pi$
$440$	18.8051	0.896499
$441$	0	0
$442$	−59.8384	−2.84622
$443$	−18.8995	−0.897941	−0.448970	−	0.893547i	$-0.648209\pi$
$443$	−18.8995	−0.897941	−0.448970	+	0.893547i	$0.648209\pi$
$444$	0	0
$445$	2.41412	0.114440
$446$	57.9096	2.74210
$447$	0	0
$448$	56.9601	2.69111
$449$	−15.7001	−0.740935	−0.370467	−	0.928845i	$-0.620803\pi$
$449$	−15.7001	−0.740935	−0.370467	+	0.928845i	$0.620803\pi$
$450$	0	0
$451$	12.5479	0.590857
$452$	28.5545	1.34309
$453$	0	0
$454$	12.5722	0.590044
$455$	−13.0530	−0.611932
$456$	0	0
$457$	9.41381	0.440359	0.220180	−	0.975459i	$-0.429336\pi$
$457$	9.41381	0.440359	0.220180	+	0.975459i	$0.429336\pi$
$458$	−20.0898	−0.938734
$459$	0	0
$460$	17.7545	0.827806
$461$	25.0721	1.16772	0.583861	−	0.811854i	$-0.301541\pi$
$461$	25.0721	1.16772	0.583861	+	0.811854i	$0.301541\pi$
$462$	0	0
$463$	28.4321	1.32135	0.660675	−	0.750672i	$-0.270270\pi$
$463$	28.4321	1.32135	0.660675	+	0.750672i	$0.270270\pi$
$464$	−126.748	−5.88413
$465$	0	0
$466$	−60.5437	−2.80463
$467$	40.0239	1.85208	0.926042	−	0.377421i	$-0.123189\pi$
$467$	40.0239	1.85208	0.926042	+	0.377421i	$0.123189\pi$
$468$	0	0
$469$	−12.9243	−0.596788
$470$	13.5272	0.623962
$471$	0	0
$472$	−92.0466	−4.23679
$473$	−9.89000	−0.454743
$474$	0	0
$475$	−28.0929	−1.28899
$476$	87.0537	3.99010
$477$	0	0
$478$	53.7153	2.45688
$479$	−39.9979	−1.82755	−0.913775	−	0.406220i	$-0.866847\pi$
$479$	−39.9979	−1.82755	−0.913775	+	0.406220i	$0.866847\pi$
$480$	0	0
$481$	−18.5251	−0.844672
$482$	−38.4599	−1.75180
$483$	0	0
$484$	−37.3528	−1.69786
$485$	−6.89605	−0.313133
$486$	0	0
$487$	−6.19213	−0.280592	−0.140296	−	0.990110i	$-0.544805\pi$
$487$	−6.19213	−0.280592	−0.140296	+	0.990110i	$0.544805\pi$
$488$	−0.0781977	−0.00353985
$489$	0	0
$490$	5.15023	0.232664
$491$	−24.1008	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$491$	−24.1008	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$492$	0	0
$493$	−57.3734	−2.58397
$494$	−79.2844	−3.56717
$495$	0	0
$496$	0	0
$497$	12.3314	0.553140
$498$	0	0
$499$	30.9325	1.38473	0.692365	−	0.721548i	$-0.256569\pi$
$499$	30.9325	1.38473	0.692365	+	0.721548i	$0.256569\pi$
$500$	51.0699	2.28391
$501$	0	0
$502$	60.1727	2.68564
$503$	−11.4990	−0.512717	−0.256358	−	0.966582i	$-0.582523\pi$
$503$	−11.4990	−0.512717	−0.256358	+	0.966582i	$0.582523\pi$
$504$	0	0
$505$	4.07491	0.181331
$506$	15.8708	0.705544
$507$	0	0
$508$	36.6519	1.62616
$509$	32.7676	1.45240	0.726200	−	0.687484i	$-0.241285\pi$
$509$	32.7676	1.45240	0.726200	+	0.687484i	$0.241285\pi$
$510$	0	0
$511$	27.6441	1.22290
$512$	6.81312	0.301100
$513$	0	0
$514$	38.6011	1.70262
$515$	11.1753	0.492443
$516$	0	0
$517$	8.72862	0.383884
$518$	37.3354	1.64042
$519$	0	0
$520$	37.8499	1.65983
$521$	26.0817	1.14266	0.571329	−	0.820721i	$-0.306428\pi$
$521$	26.0817	1.14266	0.571329	+	0.820721i	$0.306428\pi$
$522$	0	0
$523$	−21.5373	−0.941760	−0.470880	−	0.882197i	$-0.656064\pi$
$523$	−21.5373	−0.941760	−0.470880	+	0.882197i	$0.656064\pi$
$524$	50.3339	2.19885
$525$	0	0
$526$	−75.3200	−3.28411
$527$	0	0
$528$	0	0
$529$	−13.7897	−0.599553
$530$	38.1072	1.65527
$531$	0	0
$532$	115.344	5.00079
$533$	25.2557	1.09394
$534$	0	0
$535$	16.2971	0.704584
$536$	37.4767	1.61875
$537$	0	0
$538$	13.1531	0.567072
$539$	3.32327	0.143143
$540$	0	0
$541$	2.98324	0.128259	0.0641297	−	0.997942i	$-0.479573\pi$
$541$	2.98324	0.128259	0.0641297	+	0.997942i	$0.479573\pi$
$542$	12.8130	0.550366
$543$	0	0
$544$	−94.1850	−4.03815
$545$	−7.04486	−0.301769
$546$	0	0
$547$	18.1735	0.777042	0.388521	−	0.921440i	$-0.372986\pi$
$547$	18.1735	0.777042	0.388521	+	0.921440i	$0.372986\pi$
$548$	29.2007	1.24739
$549$	0	0
$550$	19.5040	0.831652
$551$	−76.0183	−3.23849
$552$	0	0
$553$	36.9294	1.57040
$554$	69.6650	2.95978
$555$	0	0
$556$	−55.1239	−2.33777
$557$	−7.64526	−0.323940	−0.161970	−	0.986796i	$-0.551785\pi$
$557$	−7.64526	−0.323940	−0.161970	+	0.986796i	$0.551785\pi$
$558$	0	0
$559$	−19.9060	−0.841935
$560$	−41.7632	−1.76482
$561$	0	0
$562$	−7.56844	−0.319256
$563$	−10.2734	−0.432973	−0.216486	−	0.976286i	$-0.569460\pi$
$563$	−10.2734	−0.432973	−0.216486	+	0.976286i	$0.569460\pi$
$564$	0	0
$565$	−6.20084	−0.260871
$566$	61.6798	2.59260
$567$	0	0
$568$	−35.7576	−1.50036
$569$	−10.4402	−0.437676	−0.218838	−	0.975761i	$-0.570227\pi$
$569$	−10.4402	−0.437676	−0.218838	+	0.975761i	$0.570227\pi$
$570$	0	0
$571$	−34.3659	−1.43817	−0.719085	−	0.694922i	$-0.755439\pi$
$571$	−34.3659	−1.43817	−0.719085	+	0.694922i	$0.755439\pi$
$572$	39.7338	1.66135
$573$	0	0
$574$	−50.9002	−2.12453
$575$	11.3187	0.472022
$576$	0	0
$577$	−7.67783	−0.319632	−0.159816	−	0.987147i	$-0.551090\pi$
$577$	−7.67783	−0.319632	−0.159816	+	0.987147i	$0.551090\pi$
$578$	−41.0777	−1.70861
$579$	0	0
$580$	59.0408	2.45154
$581$	7.68771	0.318940
$582$	0	0
$583$	24.5893	1.01838
$584$	−80.1601	−3.31705
$585$	0	0
$586$	16.8910	0.697761
$587$	−24.6559	−1.01766	−0.508829	−	0.860867i	$-0.669922\pi$
$587$	−24.6559	−1.01766	−0.508829	+	0.860867i	$0.669922\pi$
$588$	0	0
$589$	0	0
$590$	32.5193	1.33880
$591$	0	0
$592$	−59.2715	−2.43604
$593$	−11.6911	−0.480096	−0.240048	−	0.970761i	$-0.577163\pi$
$593$	−11.6911	−0.480096	−0.240048	+	0.970761i	$0.577163\pi$
$594$	0	0
$595$	−18.9044	−0.775004
$596$	−35.2133	−1.44239
$597$	0	0
$598$	31.9438	1.30628
$599$	39.3557	1.60803	0.804016	−	0.594608i	$-0.202693\pi$
$599$	39.3557	1.60803	0.804016	+	0.594608i	$0.202693\pi$
$600$	0	0
$601$	17.6769	0.721056	0.360528	−	0.932748i	$-0.382597\pi$
$601$	17.6769	0.721056	0.360528	+	0.932748i	$0.382597\pi$
$602$	40.1185	1.63511
$603$	0	0
$604$	−8.10972	−0.329980
$605$	8.11146	0.329778
$606$	0	0
$607$	3.51255	0.142570	0.0712850	−	0.997456i	$-0.477290\pi$
$607$	3.51255	0.142570	0.0712850	+	0.997456i	$0.477290\pi$
$608$	−124.793	−5.06102
$609$	0	0
$610$	0.0276266	0.00111857
$611$	17.5685	0.710744
$612$	0	0
$613$	1.10694	0.0447087	0.0223544	−	0.999750i	$-0.492884\pi$
$613$	1.10694	0.0447087	0.0223544	+	0.999750i	$0.492884\pi$
$614$	11.7250	0.473183
$615$	0	0
$616$	−49.2224	−1.98323
$617$	31.8942	1.28401	0.642007	−	0.766699i	$-0.278102\pi$
$617$	31.8942	1.28401	0.642007	+	0.766699i	$0.278102\pi$
$618$	0	0
$619$	39.8612	1.60216	0.801078	−	0.598560i	$-0.204260\pi$
$619$	39.8612	1.60216	0.801078	+	0.598560i	$0.204260\pi$
$620$	0	0
$621$	0	0
$622$	1.64139	0.0658136
$623$	−6.31897	−0.253164
$624$	0	0
$625$	7.55769	0.302308
$626$	84.4857	3.37673
$627$	0	0
$628$	103.895	4.14585
$629$	−26.8296	−1.06977
$630$	0	0
$631$	−12.0396	−0.479290	−0.239645	−	0.970861i	$-0.577031\pi$
$631$	−12.0396	−0.479290	−0.239645	+	0.970861i	$0.577031\pi$
$632$	−107.085	−4.25960
$633$	0	0
$634$	−28.7604	−1.14222
$635$	−7.95924	−0.315853
$636$	0	0
$637$	6.68887	0.265023
$638$	52.7769	2.08946
$639$	0	0
$640$	21.0054	0.830311
$641$	−26.7994	−1.05851	−0.529256	−	0.848462i	$-0.677529\pi$
$641$	−26.7994	−1.05851	−0.529256	+	0.848462i	$0.677529\pi$
$642$	0	0
$643$	17.4385	0.687707	0.343853	−	0.939023i	$-0.388268\pi$
$643$	17.4385	0.687707	0.343853	+	0.939023i	$0.388268\pi$
$644$	−46.4723	−1.83127
$645$	0	0
$646$	−114.826	−4.51778
$647$	20.6824	0.813108	0.406554	−	0.913627i	$-0.366730\pi$
$647$	20.6824	0.813108	0.406554	+	0.913627i	$0.366730\pi$
$648$	0	0
$649$	20.9836	0.823677
$650$	39.2564	1.53976
$651$	0	0
$652$	6.99069	0.273777
$653$	−4.65458	−0.182148	−0.0910739	−	0.995844i	$-0.529030\pi$
$653$	−4.65458	−0.182148	−0.0910739	+	0.995844i	$0.529030\pi$
$654$	0	0
$655$	−10.9304	−0.427087
$656$	80.8060	3.15494
$657$	0	0
$658$	−35.4074	−1.38032
$659$	−34.9124	−1.35999	−0.679997	−	0.733215i	$-0.738019\pi$
$659$	−34.9124	−1.35999	−0.679997	+	0.733215i	$0.738019\pi$
$660$	0	0
$661$	29.1668	1.13446	0.567229	−	0.823560i	$-0.308015\pi$
$661$	29.1668	1.13446	0.567229	+	0.823560i	$0.308015\pi$
$662$	9.53730	0.370678
$663$	0	0
$664$	−22.2922	−0.865104
$665$	−25.0478	−0.971314
$666$	0	0
$667$	30.6279	1.18592
$668$	33.4712	1.29504
$669$	0	0
$670$	−13.2402	−0.511514
$671$	0.0178265	0.000688184 0
$672$	0	0
$673$	−35.4056	−1.36478	−0.682392	−	0.730986i	$-0.739060\pi$
$673$	−35.4056	−1.36478	−0.682392	+	0.730986i	$0.739060\pi$
$674$	28.9495	1.11509
$675$	0	0
$676$	12.4992	0.480738
$677$	17.1093	0.657563	0.328781	−	0.944406i	$-0.393362\pi$
$677$	17.1093	0.657563	0.328781	+	0.944406i	$0.393362\pi$
$678$	0	0
$679$	18.0504	0.692711
$680$	54.8173	2.10215
$681$	0	0
$682$	0	0
$683$	20.6015	0.788296	0.394148	−	0.919047i	$-0.371040\pi$
$683$	20.6015	0.788296	0.394148	+	0.919047i	$0.371040\pi$
$684$	0	0
$685$	−6.34115	−0.242283
$686$	41.8968	1.59963
$687$	0	0
$688$	−63.6897	−2.42815
$689$	49.4919	1.88549
$690$	0	0
$691$	−43.1549	−1.64169	−0.820845	−	0.571151i	$-0.806497\pi$
$691$	−43.1549	−1.64169	−0.820845	+	0.571151i	$0.806497\pi$
$692$	−59.5686	−2.26446
$693$	0	0
$694$	22.3365	0.847881
$695$	11.9706	0.454070
$696$	0	0
$697$	36.5774	1.38547
$698$	−61.7639	−2.33780
$699$	0	0
$700$	−57.1108	−2.15858
$701$	27.1365	1.02493	0.512466	−	0.858708i	$-0.328732\pi$
$701$	27.1365	1.02493	0.512466	+	0.858708i	$0.328732\pi$
$702$	0	0
$703$	−35.5486	−1.34074
$704$	37.6528	1.41909
$705$	0	0
$706$	−23.6177	−0.888865
$707$	−10.6661	−0.401139
$708$	0	0
$709$	9.19736	0.345414	0.172707	−	0.984973i	$-0.444749\pi$
$709$	9.19736	0.345414	0.172707	+	0.984973i	$0.444749\pi$
$710$	12.6329	0.474103
$711$	0	0
$712$	18.3232	0.686692
$713$	0	0
$714$	0	0
$715$	−8.62851	−0.322688
$716$	78.8257	2.94586
$717$	0	0
$718$	−87.2762	−3.25712
$719$	−25.8892	−0.965503	−0.482751	−	0.875757i	$-0.660363\pi$
$719$	−25.8892	−0.965503	−0.482751	+	0.875757i	$0.660363\pi$
$720$	0	0
$721$	−29.2514	−1.08938
$722$	−101.194	−3.76604
$723$	0	0
$724$	31.5098	1.17105
$725$	37.6393	1.39789
$726$	0	0
$727$	24.3845	0.904370	0.452185	−	0.891924i	$-0.350645\pi$
$727$	24.3845	0.904370	0.452185	+	0.891924i	$0.350645\pi$
$728$	−99.0721	−3.67185
$729$	0	0
$730$	28.3199	1.04817
$731$	−28.8296	−1.06630
$732$	0	0
$733$	7.49203	0.276725	0.138362	−	0.990382i	$-0.455816\pi$
$733$	7.49203	0.276725	0.138362	+	0.990382i	$0.455816\pi$
$734$	18.1343	0.669351
$735$	0	0
$736$	50.2793	1.85332
$737$	−8.54346	−0.314702
$738$	0	0
$739$	43.6442	1.60548	0.802739	−	0.596330i	$-0.203375\pi$
$739$	43.6442	1.60548	0.802739	+	0.596330i	$0.203375\pi$
$740$	27.6094	1.01494
$741$	0	0
$742$	−99.7457	−3.66178
$743$	15.8274	0.580650	0.290325	−	0.956928i	$-0.406237\pi$
$743$	15.8274	0.580650	0.290325	+	0.956928i	$0.406237\pi$
$744$	0	0
$745$	7.64684	0.280159
$746$	−37.5201	−1.37371
$747$	0	0
$748$	57.5459	2.10409
$749$	−42.6576	−1.55868
$750$	0	0
$751$	34.4800	1.25819	0.629096	−	0.777328i	$-0.283425\pi$
$751$	34.4800	1.25819	0.629096	+	0.777328i	$0.283425\pi$
$752$	56.2106	2.04979
$753$	0	0
$754$	106.226	3.86853
$755$	1.76109	0.0640926
$756$	0	0
$757$	18.3192	0.665822	0.332911	−	0.942958i	$-0.391969\pi$
$757$	18.3192	0.665822	0.332911	+	0.942958i	$0.391969\pi$
$758$	−21.1857	−0.769499
$759$	0	0
$760$	72.6316	2.63463
$761$	−35.1821	−1.27535	−0.637675	−	0.770305i	$-0.720104\pi$
$761$	−35.1821	−1.27535	−0.637675	+	0.770305i	$0.720104\pi$
$762$	0	0
$763$	18.4399	0.667570
$764$	6.33160	0.229069
$765$	0	0
$766$	43.3236	1.56535
$767$	42.2345	1.52500
$768$	0	0
$769$	−14.9344	−0.538549	−0.269275	−	0.963063i	$-0.586784\pi$
$769$	−14.9344	−0.538549	−0.269275	+	0.963063i	$0.586784\pi$
$770$	17.3899	0.626687
$771$	0	0
$772$	18.6183	0.670088
$773$	−19.3964	−0.697641	−0.348821	−	0.937189i	$-0.613418\pi$
$773$	−19.3964	−0.697641	−0.348821	+	0.937189i	$0.613418\pi$
$774$	0	0
$775$	0	0
$776$	−52.3411	−1.87893
$777$	0	0
$778$	−49.2385	−1.76529
$779$	48.4641	1.73641
$780$	0	0
$781$	8.15156	0.291686
$782$	46.2637	1.65439
$783$	0	0
$784$	21.4012	0.764328
$785$	−22.5615	−0.805256
$786$	0	0
$787$	−29.0358	−1.03501	−0.517507	−	0.855679i	$-0.673140\pi$
$787$	−29.0358	−1.03501	−0.517507	+	0.855679i	$0.673140\pi$
$788$	−131.542	−4.68598
$789$	0	0
$790$	37.8321	1.34601
$791$	16.2307	0.577097
$792$	0	0
$793$	0.0358801	0.00127414
$794$	−60.2826	−2.13935
$795$	0	0
$796$	54.6098	1.93559
$797$	−24.3353	−0.862001	−0.431000	−	0.902352i	$-0.641839\pi$
$797$	−24.3353	−0.862001	−0.431000	+	0.902352i	$0.641839\pi$
$798$	0	0
$799$	25.4441	0.900148
$800$	61.7893	2.18458
$801$	0	0
$802$	27.0342	0.954610
$803$	18.2738	0.644870
$804$	0	0
$805$	10.0918	0.355690
$806$	0	0
$807$	0	0
$808$	30.9286	1.08806
$809$	−38.6992	−1.36059	−0.680296	−	0.732938i	$-0.738149\pi$
$809$	−38.6992	−1.36059	−0.680296	+	0.732938i	$0.738149\pi$
$810$	0	0
$811$	−17.5118	−0.614924	−0.307462	−	0.951560i	$-0.599480\pi$
$811$	−17.5118	−0.614924	−0.307462	+	0.951560i	$0.599480\pi$
$812$	−154.539	−5.42327
$813$	0	0
$814$	24.6802	0.865040
$815$	−1.51808	−0.0531761
$816$	0	0
$817$	−38.1985	−1.33639
$818$	−8.22830	−0.287696
$819$	0	0
$820$	−37.6404	−1.31446
$821$	2.62896	0.0917512	0.0458756	−	0.998947i	$-0.485392\pi$
$821$	2.62896	0.0917512	0.0458756	+	0.998947i	$0.485392\pi$
$822$	0	0
$823$	−36.5108	−1.27269	−0.636344	−	0.771406i	$-0.719554\pi$
$823$	−36.5108	−1.27269	−0.636344	+	0.771406i	$0.719554\pi$
$824$	84.8207	2.95487
$825$	0	0
$826$	−85.1192	−2.96168
$827$	−17.0988	−0.594583	−0.297292	−	0.954787i	$-0.596083\pi$
$827$	−17.0988	−0.594583	−0.297292	+	0.954787i	$0.596083\pi$
$828$	0	0
$829$	−52.9146	−1.83780	−0.918901	−	0.394489i	$-0.870922\pi$
$829$	−52.9146	−1.83780	−0.918901	+	0.394489i	$0.870922\pi$
$830$	7.87563	0.273367
$831$	0	0
$832$	75.7854	2.62739
$833$	9.68739	0.335648
$834$	0	0
$835$	−7.26853	−0.251538
$836$	76.2469	2.63705
$837$	0	0
$838$	52.3581	1.80868
$839$	−3.25406	−0.112343	−0.0561713	−	0.998421i	$-0.517889\pi$
$839$	−3.25406	−0.112343	−0.0561713	+	0.998421i	$0.517889\pi$
$840$	0	0
$841$	72.8504	2.51208
$842$	−12.7970	−0.441014
$843$	0	0
$844$	36.3016	1.24955
$845$	−2.71430	−0.0933747
$846$	0	0
$847$	−21.2318	−0.729532
$848$	158.350	5.43777
$849$	0	0
$850$	56.8545	1.95009
$851$	14.3226	0.490972
$852$	0	0
$853$	−28.8844	−0.988984	−0.494492	−	0.869182i	$-0.664646\pi$
$853$	−28.8844	−0.988984	−0.494492	+	0.869182i	$0.664646\pi$
$854$	−0.0723126	−0.00247449
$855$	0	0
$856$	123.695	4.22781
$857$	27.9874	0.956030	0.478015	−	0.878352i	$-0.341356\pi$
$857$	27.9874	0.956030	0.478015	+	0.878352i	$0.341356\pi$
$858$	0	0
$859$	9.90038	0.337797	0.168898	−	0.985633i	$-0.445979\pi$
$859$	9.90038	0.337797	0.168898	+	0.985633i	$0.445979\pi$
$860$	29.6674	1.01165
$861$	0	0
$862$	−53.6161	−1.82617
$863$	39.1590	1.33299	0.666494	−	0.745510i	$-0.267794\pi$
$863$	39.1590	1.33299	0.666494	+	0.745510i	$0.267794\pi$
$864$	0	0
$865$	12.9358	0.439830
$866$	85.5698	2.90778
$867$	0	0
$868$	0	0
$869$	24.4118	0.828113
$870$	0	0
$871$	−17.1958	−0.582656
$872$	−53.4705	−1.81074
$873$	0	0
$874$	61.2983	2.07345
$875$	29.0287	0.981348
$876$	0	0
$877$	−3.76703	−0.127203	−0.0636017	−	0.997975i	$-0.520259\pi$
$877$	−3.76703	−0.127203	−0.0636017	+	0.997975i	$0.520259\pi$
$878$	−12.7054	−0.428788
$879$	0	0
$880$	−27.6071	−0.930635
$881$	−8.97380	−0.302335	−0.151167	−	0.988508i	$-0.548303\pi$
$881$	−8.97380	−0.302335	−0.151167	+	0.988508i	$0.548303\pi$
$882$	0	0
$883$	−41.9265	−1.41094	−0.705470	−	0.708740i	$-0.749264\pi$
$883$	−41.9265	−1.41094	−0.705470	+	0.708740i	$0.749264\pi$
$884$	115.825	3.89561
$885$	0	0
$886$	50.6786	1.70258
$887$	−5.93349	−0.199227	−0.0996136	−	0.995026i	$-0.531761\pi$
$887$	−5.93349	−0.199227	−0.0996136	+	0.995026i	$0.531761\pi$
$888$	0	0
$889$	20.8333	0.698727
$890$	−6.47344	−0.216990
$891$	0	0
$892$	−112.092	−3.75311
$893$	33.7128	1.12816
$894$	0	0
$895$	−17.1176	−0.572179
$896$	−54.9816	−1.83681
$897$	0	0
$898$	42.0997	1.40488
$899$	0	0
$900$	0	0
$901$	71.6783	2.38795
$902$	−33.6470	−1.12032
$903$	0	0
$904$	−47.0644	−1.56534
$905$	−6.84260	−0.227456
$906$	0	0
$907$	−33.4922	−1.11209	−0.556045	−	0.831152i	$-0.687682\pi$
$907$	−33.4922	−1.11209	−0.556045	+	0.831152i	$0.687682\pi$
$908$	−24.3352	−0.807592
$909$	0	0
$910$	35.0013	1.16028
$911$	35.4191	1.17349	0.586744	−	0.809772i	$-0.300409\pi$
$911$	35.4191	1.17349	0.586744	+	0.809772i	$0.300409\pi$
$912$	0	0
$913$	5.08188	0.168186
$914$	−25.2430	−0.834964
$915$	0	0
$916$	38.8864	1.28484
$917$	28.6104	0.944797
$918$	0	0
$919$	10.7463	0.354488	0.177244	−	0.984167i	$-0.443282\pi$
$919$	10.7463	0.354488	0.177244	+	0.984167i	$0.443282\pi$
$920$	−29.2634	−0.964787
$921$	0	0
$922$	−67.2303	−2.21411
$923$	16.4070	0.540042
$924$	0	0
$925$	17.6013	0.578728
$926$	−76.2402	−2.50541
$927$	0	0
$928$	167.199	5.48858
$929$	51.9705	1.70510	0.852549	−	0.522647i	$-0.175055\pi$
$929$	51.9705	1.70510	0.852549	+	0.522647i	$0.175055\pi$
$930$	0	0
$931$	12.8355	0.420668
$932$	117.190	3.83870
$933$	0	0
$934$	−107.323	−3.51173
$935$	−12.4965	−0.408680
$936$	0	0
$937$	29.1276	0.951559	0.475779	−	0.879565i	$-0.342166\pi$
$937$	29.1276	0.951559	0.475779	+	0.879565i	$0.342166\pi$
$938$	34.6563	1.13157
$939$	0	0
$940$	−26.1836	−0.854015
$941$	15.9642	0.520419	0.260209	−	0.965552i	$-0.416208\pi$
$941$	15.9642	0.520419	0.260209	+	0.965552i	$0.416208\pi$
$942$	0	0
$943$	−19.5263	−0.635864
$944$	135.130	4.39811
$945$	0	0
$946$	26.5199	0.862236
$947$	−54.0388	−1.75603	−0.878013	−	0.478637i	$-0.841131\pi$
$947$	−54.0388	−1.75603	−0.878013	+	0.478637i	$0.841131\pi$
$948$	0	0
$949$	36.7805	1.19395
$950$	75.3308	2.44405
$951$	0	0
$952$	−143.484	−4.65036
$953$	31.4497	1.01876	0.509378	−	0.860543i	$-0.329875\pi$
$953$	31.4497	1.01876	0.509378	+	0.860543i	$0.329875\pi$
$954$	0	0
$955$	−1.37496	−0.0444925
$956$	−103.973	−3.36272
$957$	0	0
$958$	107.254	3.46521
$959$	16.5980	0.535977
$960$	0	0
$961$	0	0
$962$	49.6748	1.60158
$963$	0	0
$964$	74.4441	2.39768
$965$	−4.04311	−0.130152
$966$	0	0
$967$	−36.1032	−1.16100	−0.580500	−	0.814260i	$-0.697143\pi$
$967$	−36.1032	−1.16100	−0.580500	+	0.814260i	$0.697143\pi$
$968$	61.5661	1.97881
$969$	0	0
$970$	18.4917	0.593731
$971$	24.4315	0.784044	0.392022	−	0.919956i	$-0.371776\pi$
$971$	24.4315	0.784044	0.392022	+	0.919956i	$0.371776\pi$
$972$	0	0
$973$	−31.3330	−1.00449
$974$	16.6041	0.532030
$975$	0	0
$976$	0.114799	0.00367463
$977$	14.2775	0.456779	0.228389	−	0.973570i	$-0.426654\pi$
$977$	14.2775	0.456779	0.228389	+	0.973570i	$0.426654\pi$
$978$	0	0
$979$	−4.17709	−0.133500
$980$	−9.96894	−0.318446
$981$	0	0
$982$	64.6259	2.06230
$983$	−13.0437	−0.416030	−0.208015	−	0.978126i	$-0.566700\pi$
$983$	−13.0437	−0.416030	−0.208015	+	0.978126i	$0.566700\pi$
$984$	0	0
$985$	28.5653	0.910166
$986$	153.846	4.89945
$987$	0	0
$988$	153.465	4.88238
$989$	15.3902	0.489381
$990$	0	0
$991$	23.7517	0.754499	0.377249	−	0.926112i	$-0.376870\pi$
$991$	23.7517	0.754499	0.377249	+	0.926112i	$0.376870\pi$
$992$	0	0
$993$	0	0
$994$	−33.0666	−1.04881
$995$	−11.8589	−0.375954
$996$	0	0
$997$	12.5367	0.397042	0.198521	−	0.980097i	$-0.436386\pi$
$997$	12.5367	0.397042	0.198521	+	0.980097i	$0.436386\pi$
$998$	−82.9451	−2.62558
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bt.1.2	yes	24
3.2	odd	2	inner	8649.2.a.bt.1.23	yes	24
31.30	odd	2	inner	8649.2.a.bt.1.1	✓	24
93.92	even	2	inner	8649.2.a.bt.1.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8649.2.a.bt.1.1	✓	24	31.30	odd	2	inner
8649.2.a.bt.1.2	yes	24	1.1	even	1	trivial
8649.2.a.bt.1.23	yes	24	3.2	odd	2	inner
8649.2.a.bt.1.24	yes	24	93.92	even	2	inner

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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68149 q^{2} +5.19036 q^{4} -1.12713 q^{5} +2.95026 q^{7} -8.55491 q^{8} +O(q^{10})\)	\(q-2.68149 q^{2} +5.19036 q^{4} -1.12713 q^{5} +2.95026 q^{7} -8.55491 q^{8} +3.02238 q^{10} +1.95024 q^{11} +3.92532 q^{13} -7.91108 q^{14} +12.5591 q^{16} +5.68498 q^{17} +7.53246 q^{19} -5.85021 q^{20} -5.22953 q^{22} -3.03484 q^{23} -3.72958 q^{25} -10.5257 q^{26} +15.3129 q^{28} -10.0921 q^{29} -16.5673 q^{32} -15.2442 q^{34} -3.32532 q^{35} -4.71938 q^{37} -20.1982 q^{38} +9.64249 q^{40} +6.43404 q^{41} -5.07118 q^{43} +10.1224 q^{44} +8.13789 q^{46} +4.47567 q^{47} +1.70403 q^{49} +10.0008 q^{50} +20.3738 q^{52} +12.6084 q^{53} -2.19817 q^{55} -25.2392 q^{56} +27.0618 q^{58} +10.7595 q^{59} +0.00914068 q^{61} +19.3068 q^{64} -4.42434 q^{65} -4.38073 q^{67} +29.5071 q^{68} +8.91680 q^{70} +4.17978 q^{71} +9.37006 q^{73} +12.6550 q^{74} +39.0962 q^{76} +5.75370 q^{77} +12.5173 q^{79} -14.1558 q^{80} -17.2528 q^{82} +2.60577 q^{83} -6.40770 q^{85} +13.5983 q^{86} -16.6841 q^{88} -2.14184 q^{89} +11.5807 q^{91} -15.7519 q^{92} -12.0014 q^{94} -8.49005 q^{95} +6.11825 q^{97} -4.56934 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7}+O(q^{10}) \) 24 * q + 32 * q^4 + 16 * q^7	\( 24 q + 32 q^{4} + 16 q^{7} + 40 q^{10} + 64 q^{16} + 32 q^{19} + 24 q^{25} + 104 q^{28} + 160 q^{40} - 40 q^{49} + 136 q^{64} + 80 q^{67} + 72 q^{70} + 120 q^{76} - 152 q^{82} + 80 q^{94} + 32 q^{97}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 40 * q^10 + 64 * q^16 + 32 * q^19 + 24 * q^25 + 104 * q^28 + 160 * q^40 - 40 * q^49 + 136 * q^64 + 80 * q^67 + 72 * q^70 + 120 * q^76 - 152 * q^82 + 80 * q^94 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more