LMFDB - Embedded newform 6044.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, 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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26383 q^{3} -4.12384 q^{5} -1.32793 q^{7} +7.65258 q^{9} +O(q^{10})$	$q-3.26383 q^{3} -4.12384 q^{5} -1.32793 q^{7} +7.65258 q^{9} +0.376598 q^{11} +0.616508 q^{13} +13.4595 q^{15} -6.25567 q^{17} +3.14876 q^{19} +4.33415 q^{21} -1.53701 q^{23} +12.0060 q^{25} -15.1852 q^{27} +8.05021 q^{29} +1.93326 q^{31} -1.22915 q^{33} +5.47618 q^{35} +1.27308 q^{37} -2.01218 q^{39} -9.43783 q^{41} +2.35398 q^{43} -31.5580 q^{45} -5.73950 q^{47} -5.23659 q^{49} +20.4174 q^{51} -10.5402 q^{53} -1.55303 q^{55} -10.2770 q^{57} -6.91047 q^{59} -3.87704 q^{61} -10.1621 q^{63} -2.54238 q^{65} +12.4397 q^{67} +5.01653 q^{69} -13.0020 q^{71} -9.57463 q^{73} -39.1856 q^{75} -0.500098 q^{77} +3.82338 q^{79} +26.6043 q^{81} -7.39807 q^{83} +25.7974 q^{85} -26.2745 q^{87} +7.48893 q^{89} -0.818682 q^{91} -6.30984 q^{93} -12.9850 q^{95} -9.67930 q^{97} +2.88195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−3.26383	−1.88437	−0.942186	−	0.335089i	$-0.891233\pi$
$3$	−3.26383	−1.88437	−0.942186	+	0.335089i	$0.891233\pi$
$4$	0	0
$5$	−4.12384	−1.84424	−0.922118	−	0.386909i	$-0.873543\pi$
$5$	−4.12384	−1.84424	−0.922118	+	0.386909i	$0.873543\pi$
$6$	0	0
$7$	−1.32793	−0.501912	−0.250956	−	0.967999i	$-0.580745\pi$
$7$	−1.32793	−0.501912	−0.250956	+	0.967999i	$0.580745\pi$
$8$	0	0
$9$	7.65258	2.55086
$10$	0	0
$11$	0.376598	0.113549	0.0567743	−	0.998387i	$-0.481918\pi$
$11$	0.376598	0.113549	0.0567743	+	0.998387i	$0.481918\pi$
$12$	0	0
$13$	0.616508	0.170989	0.0854943	−	0.996339i	$-0.472753\pi$
$13$	0.616508	0.170989	0.0854943	+	0.996339i	$0.472753\pi$
$14$	0	0
$15$	13.4595	3.47523
$16$	0	0
$17$	−6.25567	−1.51722	−0.758611	−	0.651544i	$-0.774122\pi$
$17$	−6.25567	−1.51722	−0.758611	+	0.651544i	$0.774122\pi$
$18$	0	0
$19$	3.14876	0.722374	0.361187	−	0.932493i	$-0.382372\pi$
$19$	3.14876	0.722374	0.361187	+	0.932493i	$0.382372\pi$
$20$	0	0
$21$	4.33415	0.945789
$22$	0	0
$23$	−1.53701	−0.320488	−0.160244	−	0.987077i	$-0.551228\pi$
$23$	−1.53701	−0.320488	−0.160244	+	0.987077i	$0.551228\pi$
$24$	0	0
$25$	12.0060	2.40121
$26$	0	0
$27$	−15.1852	−2.92240
$28$	0	0
$29$	8.05021	1.49489	0.747444	−	0.664325i	$-0.231281\pi$
$29$	8.05021	1.49489	0.747444	+	0.664325i	$0.231281\pi$
$30$	0	0
$31$	1.93326	0.347224	0.173612	−	0.984814i	$-0.444456\pi$
$31$	1.93326	0.347224	0.173612	+	0.984814i	$0.444456\pi$
$32$	0	0
$33$	−1.22915	−0.213968
$34$	0	0
$35$	5.47618	0.925644
$36$	0	0
$37$	1.27308	0.209293	0.104646	−	0.994510i	$-0.466629\pi$
$37$	1.27308	0.209293	0.104646	+	0.994510i	$0.466629\pi$
$38$	0	0
$39$	−2.01218	−0.322206
$40$	0	0
$41$	−9.43783	−1.47394	−0.736971	−	0.675925i	$-0.763744\pi$
$41$	−9.43783	−1.47394	−0.736971	+	0.675925i	$0.763744\pi$
$42$	0	0
$43$	2.35398	0.358979	0.179490	−	0.983760i	$-0.442555\pi$
$43$	2.35398	0.358979	0.179490	+	0.983760i	$0.442555\pi$
$44$	0	0
$45$	−31.5580	−4.70439
$46$	0	0
$47$	−5.73950	−0.837192	−0.418596	−	0.908173i	$-0.637478\pi$
$47$	−5.73950	−0.837192	−0.418596	+	0.908173i	$0.637478\pi$
$48$	0	0
$49$	−5.23659	−0.748085
$50$	0	0
$51$	20.4174	2.85901
$52$	0	0
$53$	−10.5402	−1.44781	−0.723905	−	0.689900i	$-0.757655\pi$
$53$	−10.5402	−1.44781	−0.723905	+	0.689900i	$0.757655\pi$
$54$	0	0
$55$	−1.55303	−0.209411
$56$	0	0
$57$	−10.2770	−1.36122
$58$	0	0
$59$	−6.91047	−0.899666	−0.449833	−	0.893113i	$-0.648517\pi$
$59$	−6.91047	−0.899666	−0.449833	+	0.893113i	$0.648517\pi$
$60$	0	0
$61$	−3.87704	−0.496404	−0.248202	−	0.968708i	$-0.579840\pi$
$61$	−3.87704	−0.496404	−0.248202	+	0.968708i	$0.579840\pi$
$62$	0	0
$63$	−10.1621	−1.28031
$64$	0	0
$65$	−2.54238	−0.315343
$66$	0	0
$67$	12.4397	1.51975	0.759876	−	0.650068i	$-0.225260\pi$
$67$	12.4397	1.51975	0.759876	+	0.650068i	$0.225260\pi$
$68$	0	0
$69$	5.01653	0.603919
$70$	0	0
$71$	−13.0020	−1.54305	−0.771526	−	0.636198i	$-0.780506\pi$
$71$	−13.0020	−1.54305	−0.771526	+	0.636198i	$0.780506\pi$
$72$	0	0
$73$	−9.57463	−1.12063	−0.560313	−	0.828281i	$-0.689319\pi$
$73$	−9.57463	−1.12063	−0.560313	+	0.828281i	$0.689319\pi$
$74$	0	0
$75$	−39.1856	−4.52477
$76$	0	0
$77$	−0.500098	−0.0569914
$78$	0	0
$79$	3.82338	0.430163	0.215082	−	0.976596i	$-0.430998\pi$
$79$	3.82338	0.430163	0.215082	+	0.976596i	$0.430998\pi$
$80$	0	0
$81$	26.6043	2.95603
$82$	0	0
$83$	−7.39807	−0.812044	−0.406022	−	0.913863i	$-0.633084\pi$
$83$	−7.39807	−0.812044	−0.406022	+	0.913863i	$0.633084\pi$
$84$	0	0
$85$	25.7974	2.79812
$86$	0	0
$87$	−26.2745	−2.81692
$88$	0	0
$89$	7.48893	0.793825	0.396913	−	0.917856i	$-0.370082\pi$
$89$	7.48893	0.793825	0.396913	+	0.917856i	$0.370082\pi$
$90$	0	0
$91$	−0.818682	−0.0858212
$92$	0	0
$93$	−6.30984	−0.654300
$94$	0	0
$95$	−12.9850	−1.33223
$96$	0	0
$97$	−9.67930	−0.982784	−0.491392	−	0.870939i	$-0.663512\pi$
$97$	−9.67930	−0.982784	−0.491392	+	0.870939i	$0.663512\pi$
$98$	0	0
$99$	2.88195	0.289647
$100$	0	0
$101$	12.7002	1.26372	0.631861	−	0.775082i	$-0.282291\pi$
$101$	12.7002	1.26372	0.631861	+	0.775082i	$0.282291\pi$
$102$	0	0
$103$	3.54233	0.349036	0.174518	−	0.984654i	$-0.444163\pi$
$103$	3.54233	0.349036	0.174518	+	0.984654i	$0.444163\pi$
$104$	0	0
$105$	−17.8733	−1.74426
$106$	0	0
$107$	−2.20848	−0.213501	−0.106751	−	0.994286i	$-0.534045\pi$
$107$	−2.20848	−0.213501	−0.106751	+	0.994286i	$0.534045\pi$
$108$	0	0
$109$	−1.87495	−0.179588	−0.0897939	−	0.995960i	$-0.528621\pi$
$109$	−1.87495	−0.179588	−0.0897939	+	0.995960i	$0.528621\pi$
$110$	0	0
$111$	−4.15511	−0.394385
$112$	0	0
$113$	−18.3119	−1.72264	−0.861319	−	0.508065i	$-0.830361\pi$
$113$	−18.3119	−1.72264	−0.861319	+	0.508065i	$0.830361\pi$
$114$	0	0
$115$	6.33836	0.591055
$116$	0	0
$117$	4.71788	0.436168
$118$	0	0
$119$	8.30711	0.761512
$120$	0	0
$121$	−10.8582	−0.987107
$122$	0	0
$123$	30.8035	2.77746
$124$	0	0
$125$	−28.8917	−2.58415
$126$	0	0
$127$	9.02253	0.800620	0.400310	−	0.916380i	$-0.368902\pi$
$127$	9.02253	0.800620	0.400310	+	0.916380i	$0.368902\pi$
$128$	0	0
$129$	−7.68300	−0.676450
$130$	0	0
$131$	2.50311	0.218698	0.109349	−	0.994003i	$-0.465123\pi$
$131$	2.50311	0.218698	0.109349	+	0.994003i	$0.465123\pi$
$132$	0	0
$133$	−4.18134	−0.362568
$134$	0	0
$135$	62.6214	5.38959
$136$	0	0
$137$	1.59744	0.136479	0.0682393	−	0.997669i	$-0.478262\pi$
$137$	1.59744	0.136479	0.0682393	+	0.997669i	$0.478262\pi$
$138$	0	0
$139$	7.17016	0.608165	0.304082	−	0.952646i	$-0.401650\pi$
$139$	7.17016	0.608165	0.304082	+	0.952646i	$0.401650\pi$
$140$	0	0
$141$	18.7327	1.57758
$142$	0	0
$143$	0.232176	0.0194155
$144$	0	0
$145$	−33.1978	−2.75692
$146$	0	0
$147$	17.0913	1.40967
$148$	0	0
$149$	4.53490	0.371514	0.185757	−	0.982596i	$-0.440526\pi$
$149$	4.53490	0.371514	0.185757	+	0.982596i	$0.440526\pi$
$150$	0	0
$151$	−15.7241	−1.27961	−0.639805	−	0.768538i	$-0.720985\pi$
$151$	−15.7241	−1.27961	−0.639805	+	0.768538i	$0.720985\pi$
$152$	0	0
$153$	−47.8720	−3.87022
$154$	0	0
$155$	−7.97246	−0.640363
$156$	0	0
$157$	−1.57816	−0.125951	−0.0629755	−	0.998015i	$-0.520059\pi$
$157$	−1.57816	−0.125951	−0.0629755	+	0.998015i	$0.520059\pi$
$158$	0	0
$159$	34.4015	2.72821
$160$	0	0
$161$	2.04104	0.160857
$162$	0	0
$163$	−19.7211	−1.54468	−0.772340	−	0.635210i	$-0.780914\pi$
$163$	−19.7211	−1.54468	−0.772340	+	0.635210i	$0.780914\pi$
$164$	0	0
$165$	5.06882	0.394607
$166$	0	0
$167$	−13.7291	−1.06239	−0.531196	−	0.847249i	$-0.678257\pi$
$167$	−13.7291	−1.06239	−0.531196	+	0.847249i	$0.678257\pi$
$168$	0	0
$169$	−12.6199	−0.970763
$170$	0	0
$171$	24.0961	1.84268
$172$	0	0
$173$	2.45581	0.186712	0.0933559	−	0.995633i	$-0.470241\pi$
$173$	2.45581	0.186712	0.0933559	+	0.995633i	$0.470241\pi$
$174$	0	0
$175$	−15.9432	−1.20519
$176$	0	0
$177$	22.5546	1.69531
$178$	0	0
$179$	23.5677	1.76153	0.880765	−	0.473553i	$-0.157029\pi$
$179$	23.5677	1.76153	0.880765	+	0.473553i	$0.157029\pi$
$180$	0	0
$181$	−9.24417	−0.687114	−0.343557	−	0.939132i	$-0.611632\pi$
$181$	−9.24417	−0.687114	−0.343557	+	0.939132i	$0.611632\pi$
$182$	0	0
$183$	12.6540	0.935410
$184$	0	0
$185$	−5.24996	−0.385985
$186$	0	0
$187$	−2.35587	−0.172279
$188$	0	0
$189$	20.1650	1.46679
$190$	0	0
$191$	−3.21885	−0.232908	−0.116454	−	0.993196i	$-0.537153\pi$
$191$	−3.21885	−0.232908	−0.116454	+	0.993196i	$0.537153\pi$
$192$	0	0
$193$	17.3477	1.24871	0.624357	−	0.781139i	$-0.285361\pi$
$193$	17.3477	1.24871	0.624357	+	0.781139i	$0.285361\pi$
$194$	0	0
$195$	8.29789	0.594224
$196$	0	0
$197$	−22.0713	−1.57252	−0.786259	−	0.617897i	$-0.787985\pi$
$197$	−22.0713	−1.57252	−0.786259	+	0.617897i	$0.787985\pi$
$198$	0	0
$199$	−16.2726	−1.15353	−0.576766	−	0.816910i	$-0.695685\pi$
$199$	−16.2726	−1.15353	−0.576766	+	0.816910i	$0.695685\pi$
$200$	0	0
$201$	−40.6011	−2.86378
$202$	0	0
$203$	−10.6902	−0.750302
$204$	0	0
$205$	38.9201	2.71830
$206$	0	0
$207$	−11.7621	−0.817520
$208$	0	0
$209$	1.18582	0.0820247
$210$	0	0
$211$	−1.51943	−0.104602	−0.0523009	−	0.998631i	$-0.516655\pi$
$211$	−1.51943	−0.104602	−0.0523009	+	0.998631i	$0.516655\pi$
$212$	0	0
$213$	42.4363	2.90768
$214$	0	0
$215$	−9.70744	−0.662042
$216$	0	0
$217$	−2.56725	−0.174276
$218$	0	0
$219$	31.2500	2.11168
$220$	0	0
$221$	−3.85667	−0.259428
$222$	0	0
$223$	1.68359	0.112742	0.0563708	−	0.998410i	$-0.482047\pi$
$223$	1.68359	0.112742	0.0563708	+	0.998410i	$0.482047\pi$
$224$	0	0
$225$	91.8771	6.12514
$226$	0	0
$227$	28.1683	1.86960	0.934799	−	0.355177i	$-0.115579\pi$
$227$	28.1683	1.86960	0.934799	+	0.355177i	$0.115579\pi$
$228$	0	0
$229$	−8.01506	−0.529650	−0.264825	−	0.964296i	$-0.585314\pi$
$229$	−8.01506	−0.529650	−0.264825	+	0.964296i	$0.585314\pi$
$230$	0	0
$231$	1.63223	0.107393
$232$	0	0
$233$	−22.0667	−1.44564	−0.722819	−	0.691038i	$-0.757154\pi$
$233$	−22.0667	−1.44564	−0.722819	+	0.691038i	$0.757154\pi$
$234$	0	0
$235$	23.6688	1.54398
$236$	0	0
$237$	−12.4788	−0.810588
$238$	0	0
$239$	15.8293	1.02391	0.511955	−	0.859012i	$-0.328921\pi$
$239$	15.8293	1.02391	0.511955	+	0.859012i	$0.328921\pi$
$240$	0	0
$241$	20.9167	1.34737	0.673683	−	0.739020i	$-0.264711\pi$
$241$	20.9167	1.34737	0.673683	+	0.739020i	$0.264711\pi$
$242$	0	0
$243$	−41.2761	−2.64786
$244$	0	0
$245$	21.5948	1.37964
$246$	0	0
$247$	1.94123	0.123518
$248$	0	0
$249$	24.1460	1.53019
$250$	0	0
$251$	−3.66428	−0.231287	−0.115644	−	0.993291i	$-0.536893\pi$
$251$	−3.66428	−0.231287	−0.115644	+	0.993291i	$0.536893\pi$
$252$	0	0
$253$	−0.578834	−0.0363910
$254$	0	0
$255$	−84.1981	−5.27269
$256$	0	0
$257$	−2.47579	−0.154436	−0.0772178	−	0.997014i	$-0.524604\pi$
$257$	−2.47579	−0.154436	−0.0772178	+	0.997014i	$0.524604\pi$
$258$	0	0
$259$	−1.69056	−0.105046
$260$	0	0
$261$	61.6049	3.81325
$262$	0	0
$263$	8.21937	0.506828	0.253414	−	0.967358i	$-0.418447\pi$
$263$	8.21937	0.506828	0.253414	+	0.967358i	$0.418447\pi$
$264$	0	0
$265$	43.4661	2.67010
$266$	0	0
$267$	−24.4426	−1.49586
$268$	0	0
$269$	−0.620028	−0.0378038	−0.0189019	−	0.999821i	$-0.506017\pi$
$269$	−0.620028	−0.0378038	−0.0189019	+	0.999821i	$0.506017\pi$
$270$	0	0
$271$	−5.66667	−0.344226	−0.172113	−	0.985077i	$-0.555059\pi$
$271$	−5.66667	−0.344226	−0.172113	+	0.985077i	$0.555059\pi$
$272$	0	0
$273$	2.67204	0.161719
$274$	0	0
$275$	4.52145	0.272654
$276$	0	0
$277$	−10.8666	−0.652909	−0.326454	−	0.945213i	$-0.605854\pi$
$277$	−10.8666	−0.652909	−0.326454	+	0.945213i	$0.605854\pi$
$278$	0	0
$279$	14.7945	0.885721
$280$	0	0
$281$	−18.1731	−1.08411	−0.542057	−	0.840342i	$-0.682354\pi$
$281$	−18.1731	−1.08411	−0.542057	+	0.840342i	$0.682354\pi$
$282$	0	0
$283$	23.0067	1.36760	0.683802	−	0.729668i	$-0.260325\pi$
$283$	23.0067	1.36760	0.683802	+	0.729668i	$0.260325\pi$
$284$	0	0
$285$	42.3807	2.51042
$286$	0	0
$287$	12.5328	0.739789
$288$	0	0
$289$	22.1334	1.30196
$290$	0	0
$291$	31.5916	1.85193
$292$	0	0
$293$	10.7401	0.627445	0.313722	−	0.949515i	$-0.398424\pi$
$293$	10.7401	0.627445	0.313722	+	0.949515i	$0.398424\pi$
$294$	0	0
$295$	28.4976	1.65920
$296$	0	0
$297$	−5.71873	−0.331835
$298$	0	0
$299$	−0.947577	−0.0547998
$300$	0	0
$301$	−3.12593	−0.180176
$302$	0	0
$303$	−41.4514	−2.38132
$304$	0	0
$305$	15.9883	0.915486
$306$	0	0
$307$	−8.58077	−0.489731	−0.244865	−	0.969557i	$-0.578744\pi$
$307$	−8.58077	−0.489731	−0.244865	+	0.969557i	$0.578744\pi$
$308$	0	0
$309$	−11.5616	−0.657714
$310$	0	0
$311$	29.0525	1.64742	0.823708	−	0.567015i	$-0.191902\pi$
$311$	29.0525	1.64742	0.823708	+	0.567015i	$0.191902\pi$
$312$	0	0
$313$	−11.7341	−0.663251	−0.331625	−	0.943411i	$-0.607597\pi$
$313$	−11.7341	−0.663251	−0.331625	+	0.943411i	$0.607597\pi$
$314$	0	0
$315$	41.9069	2.36119
$316$	0	0
$317$	9.31432	0.523144	0.261572	−	0.965184i	$-0.415759\pi$
$317$	9.31432	0.523144	0.261572	+	0.965184i	$0.415759\pi$
$318$	0	0
$319$	3.03170	0.169742
$320$	0	0
$321$	7.20809	0.402316
$322$	0	0
$323$	−19.6976	−1.09600
$324$	0	0
$325$	7.40181	0.410579
$326$	0	0
$327$	6.11952	0.338410
$328$	0	0
$329$	7.62168	0.420197
$330$	0	0
$331$	−5.94789	−0.326926	−0.163463	−	0.986549i	$-0.552266\pi$
$331$	−5.94789	−0.326926	−0.163463	+	0.986549i	$0.552266\pi$
$332$	0	0
$333$	9.74233	0.533876
$334$	0	0
$335$	−51.2993	−2.80278
$336$	0	0
$337$	21.1389	1.15151	0.575755	−	0.817622i	$-0.304708\pi$
$337$	21.1389	1.15151	0.575755	+	0.817622i	$0.304708\pi$
$338$	0	0
$339$	59.7669	3.24609
$340$	0	0
$341$	0.728064	0.0394269
$342$	0	0
$343$	16.2494	0.877384
$344$	0	0
$345$	−20.6873	−1.11377
$346$	0	0
$347$	0.439987	0.0236197	0.0118099	−	0.999930i	$-0.496241\pi$
$347$	0.439987	0.0236197	0.0118099	+	0.999930i	$0.496241\pi$
$348$	0	0
$349$	10.5812	0.566397	0.283199	−	0.959061i	$-0.408604\pi$
$349$	10.5812	0.566397	0.283199	+	0.959061i	$0.408604\pi$
$350$	0	0
$351$	−9.36182	−0.499697
$352$	0	0
$353$	24.1472	1.28523	0.642613	−	0.766191i	$-0.277850\pi$
$353$	24.1472	1.28523	0.642613	+	0.766191i	$0.277850\pi$
$354$	0	0
$355$	53.6181	2.84575
$356$	0	0
$357$	−27.1130	−1.43497
$358$	0	0
$359$	15.2153	0.803033	0.401516	−	0.915852i	$-0.368483\pi$
$359$	15.2153	0.803033	0.401516	+	0.915852i	$0.368483\pi$
$360$	0	0
$361$	−9.08533	−0.478175
$362$	0	0
$363$	35.4392	1.86008
$364$	0	0
$365$	39.4842	2.06670
$366$	0	0
$367$	9.04056	0.471914	0.235957	−	0.971764i	$-0.424178\pi$
$367$	9.04056	0.471914	0.235957	+	0.971764i	$0.424178\pi$
$368$	0	0
$369$	−72.2238	−3.75982
$370$	0	0
$371$	13.9967	0.726673
$372$	0	0
$373$	16.8745	0.873731	0.436866	−	0.899527i	$-0.356089\pi$
$373$	16.8745	0.873731	0.436866	+	0.899527i	$0.356089\pi$
$374$	0	0
$375$	94.2976	4.86951
$376$	0	0
$377$	4.96302	0.255609
$378$	0	0
$379$	18.5865	0.954724	0.477362	−	0.878707i	$-0.341593\pi$
$379$	18.5865	0.954724	0.477362	+	0.878707i	$0.341593\pi$
$380$	0	0
$381$	−29.4480	−1.50867
$382$	0	0
$383$	−36.0026	−1.83965	−0.919824	−	0.392332i	$-0.871668\pi$
$383$	−36.0026	−1.83965	−0.919824	+	0.392332i	$0.871668\pi$
$384$	0	0
$385$	2.06232	0.105106
$386$	0	0
$387$	18.0140	0.915705
$388$	0	0
$389$	−31.1731	−1.58054	−0.790269	−	0.612760i	$-0.790059\pi$
$389$	−31.1731	−1.58054	−0.790269	+	0.612760i	$0.790059\pi$
$390$	0	0
$391$	9.61500	0.486251
$392$	0	0
$393$	−8.16973	−0.412108
$394$	0	0
$395$	−15.7670	−0.793323
$396$	0	0
$397$	−29.6222	−1.48669	−0.743347	−	0.668906i	$-0.766763\pi$
$397$	−29.6222	−1.48669	−0.743347	+	0.668906i	$0.766763\pi$
$398$	0	0
$399$	13.6472	0.683214
$400$	0	0
$401$	−12.3981	−0.619133	−0.309566	−	0.950878i	$-0.600184\pi$
$401$	−12.3981	−0.619133	−0.309566	+	0.950878i	$0.600184\pi$
$402$	0	0
$403$	1.19187	0.0593714
$404$	0	0
$405$	−109.712	−5.45161
$406$	0	0
$407$	0.479439	0.0237649
$408$	0	0
$409$	−17.2935	−0.855109	−0.427554	−	0.903990i	$-0.640625\pi$
$409$	−17.2935	−0.855109	−0.427554	+	0.903990i	$0.640625\pi$
$410$	0	0
$411$	−5.21378	−0.257177
$412$	0	0
$413$	9.17664	0.451553
$414$	0	0
$415$	30.5084	1.49760
$416$	0	0
$417$	−23.4022	−1.14601
$418$	0	0
$419$	5.20962	0.254506	0.127253	−	0.991870i	$-0.459384\pi$
$419$	5.20962	0.254506	0.127253	+	0.991870i	$0.459384\pi$
$420$	0	0
$421$	−36.9461	−1.80064	−0.900322	−	0.435224i	$-0.856669\pi$
$421$	−36.9461	−1.80064	−0.900322	+	0.435224i	$0.856669\pi$
$422$	0	0
$423$	−43.9220	−2.13556
$424$	0	0
$425$	−75.1057	−3.64316
$426$	0	0
$427$	5.14845	0.249151
$428$	0	0
$429$	−0.757783	−0.0365861
$430$	0	0
$431$	−5.30300	−0.255436	−0.127718	−	0.991810i	$-0.540765\pi$
$431$	−5.30300	−0.255436	−0.127718	+	0.991810i	$0.540765\pi$
$432$	0	0
$433$	30.6503	1.47296	0.736481	−	0.676458i	$-0.236486\pi$
$433$	30.6503	1.47296	0.736481	+	0.676458i	$0.236486\pi$
$434$	0	0
$435$	108.352	5.19507
$436$	0	0
$437$	−4.83966	−0.231512
$438$	0	0
$439$	2.00394	0.0956427	0.0478213	−	0.998856i	$-0.484772\pi$
$439$	2.00394	0.0956427	0.0478213	+	0.998856i	$0.484772\pi$
$440$	0	0
$441$	−40.0734	−1.90826
$442$	0	0
$443$	12.7528	0.605902	0.302951	−	0.953006i	$-0.402028\pi$
$443$	12.7528	0.605902	0.302951	+	0.953006i	$0.402028\pi$
$444$	0	0
$445$	−30.8831	−1.46400
$446$	0	0
$447$	−14.8012	−0.700071
$448$	0	0
$449$	16.1824	0.763695	0.381847	−	0.924225i	$-0.375288\pi$
$449$	16.1824	0.763695	0.381847	+	0.924225i	$0.375288\pi$
$450$	0	0
$451$	−3.55427	−0.167364
$452$	0	0
$453$	51.3208	2.41126
$454$	0	0
$455$	3.37611	0.158275
$456$	0	0
$457$	−41.6336	−1.94754	−0.973768	−	0.227544i	$-0.926931\pi$
$457$	−41.6336	−1.94754	−0.973768	+	0.227544i	$0.926931\pi$
$458$	0	0
$459$	94.9938	4.43393
$460$	0	0
$461$	25.8383	1.20341	0.601704	−	0.798719i	$-0.294489\pi$
$461$	25.8383	1.20341	0.601704	+	0.798719i	$0.294489\pi$
$462$	0	0
$463$	33.8044	1.57102	0.785511	−	0.618847i	$-0.212400\pi$
$463$	33.8044	1.57102	0.785511	+	0.618847i	$0.212400\pi$
$464$	0	0
$465$	26.0207	1.20668
$466$	0	0
$467$	−24.9029	−1.15237	−0.576184	−	0.817320i	$-0.695459\pi$
$467$	−24.9029	−1.15237	−0.576184	+	0.817320i	$0.695459\pi$
$468$	0	0
$469$	−16.5191	−0.762782
$470$	0	0
$471$	5.15085	0.237339
$472$	0	0
$473$	0.886506	0.0407616
$474$	0	0
$475$	37.8041	1.73457
$476$	0	0
$477$	−80.6599	−3.69316
$478$	0	0
$479$	34.4045	1.57198	0.785991	−	0.618237i	$-0.212153\pi$
$479$	34.4045	1.57198	0.785991	+	0.618237i	$0.212153\pi$
$480$	0	0
$481$	0.784863	0.0357867
$482$	0	0
$483$	−6.66161	−0.303114
$484$	0	0
$485$	39.9158	1.81248
$486$	0	0
$487$	21.7362	0.984961	0.492480	−	0.870324i	$-0.336090\pi$
$487$	21.7362	0.984961	0.492480	+	0.870324i	$0.336090\pi$
$488$	0	0
$489$	64.3664	2.91075
$490$	0	0
$491$	−7.07744	−0.319401	−0.159700	−	0.987166i	$-0.551053\pi$
$491$	−7.07744	−0.319401	−0.159700	+	0.987166i	$0.551053\pi$
$492$	0	0
$493$	−50.3595	−2.26808
$494$	0	0
$495$	−11.8847	−0.534177
$496$	0	0
$497$	17.2658	0.774476
$498$	0	0
$499$	13.3134	0.595988	0.297994	−	0.954568i	$-0.403682\pi$
$499$	13.3134	0.595988	0.297994	+	0.954568i	$0.403682\pi$
$500$	0	0
$501$	44.8095	2.00194
$502$	0	0
$503$	−25.7392	−1.14766	−0.573828	−	0.818976i	$-0.694542\pi$
$503$	−25.7392	−1.14766	−0.573828	+	0.818976i	$0.694542\pi$
$504$	0	0
$505$	−52.3737	−2.33060
$506$	0	0
$507$	41.1893	1.82928
$508$	0	0
$509$	13.1228	0.581660	0.290830	−	0.956775i	$-0.406069\pi$
$509$	13.1228	0.581660	0.290830	+	0.956775i	$0.406069\pi$
$510$	0	0
$511$	12.7145	0.562455
$512$	0	0
$513$	−47.8146	−2.11107
$514$	0	0
$515$	−14.6080	−0.643705
$516$	0	0
$517$	−2.16149	−0.0950620
$518$	0	0
$519$	−8.01535	−0.351835
$520$	0	0
$521$	31.5969	1.38429	0.692143	−	0.721761i	$-0.256667\pi$
$521$	31.5969	1.38429	0.692143	+	0.721761i	$0.256667\pi$
$522$	0	0
$523$	18.4730	0.807768	0.403884	−	0.914810i	$-0.367660\pi$
$523$	18.4730	0.807768	0.403884	+	0.914810i	$0.367660\pi$
$524$	0	0
$525$	52.0359	2.27103
$526$	0	0
$527$	−12.0939	−0.526816
$528$	0	0
$529$	−20.6376	−0.897288
$530$	0	0
$531$	−52.8829	−2.29492
$532$	0	0
$533$	−5.81850	−0.252027
$534$	0	0
$535$	9.10739	0.393747
$536$	0	0
$537$	−76.9208	−3.31938
$538$	0	0
$539$	−1.97209	−0.0849440
$540$	0	0
$541$	−44.8452	−1.92805	−0.964024	−	0.265816i	$-0.914359\pi$
$541$	−44.8452	−1.92805	−0.964024	+	0.265816i	$0.914359\pi$
$542$	0	0
$543$	30.1714	1.29478
$544$	0	0
$545$	7.73199	0.331202
$546$	0	0
$547$	12.7173	0.543755	0.271877	−	0.962332i	$-0.412356\pi$
$547$	12.7173	0.543755	0.271877	+	0.962332i	$0.412356\pi$
$548$	0	0
$549$	−29.6694	−1.26626
$550$	0	0
$551$	25.3482	1.07987
$552$	0	0
$553$	−5.07719	−0.215904
$554$	0	0
$555$	17.1350	0.727339
$556$	0	0
$557$	−4.74097	−0.200881	−0.100441	−	0.994943i	$-0.532025\pi$
$557$	−4.74097	−0.200881	−0.100441	+	0.994943i	$0.532025\pi$
$558$	0	0
$559$	1.45125	0.0613813
$560$	0	0
$561$	7.68917	0.324637
$562$	0	0
$563$	−26.6026	−1.12116	−0.560582	−	0.828099i	$-0.689423\pi$
$563$	−26.6026	−1.12116	−0.560582	+	0.828099i	$0.689423\pi$
$564$	0	0
$565$	75.5152	3.17695
$566$	0	0
$567$	−35.3287	−1.48367
$568$	0	0
$569$	2.48512	0.104182	0.0520909	−	0.998642i	$-0.483411\pi$
$569$	2.48512	0.104182	0.0520909	+	0.998642i	$0.483411\pi$
$570$	0	0
$571$	−29.8478	−1.24909	−0.624546	−	0.780988i	$-0.714716\pi$
$571$	−29.8478	−1.24909	−0.624546	+	0.780988i	$0.714716\pi$
$572$	0	0
$573$	10.5058	0.438885
$574$	0	0
$575$	−18.4533	−0.769557
$576$	0	0
$577$	−1.89023	−0.0786914	−0.0393457	−	0.999226i	$-0.512527\pi$
$577$	−1.89023	−0.0786914	−0.0393457	+	0.999226i	$0.512527\pi$
$578$	0	0
$579$	−56.6199	−2.35304
$580$	0	0
$581$	9.82415	0.407574
$582$	0	0
$583$	−3.96943	−0.164397
$584$	0	0
$585$	−19.4558	−0.804397
$586$	0	0
$587$	4.33222	0.178810	0.0894050	−	0.995995i	$-0.471503\pi$
$587$	4.33222	0.178810	0.0894050	+	0.995995i	$0.471503\pi$
$588$	0	0
$589$	6.08737	0.250826
$590$	0	0
$591$	72.0371	2.96321
$592$	0	0
$593$	3.40453	0.139807	0.0699036	−	0.997554i	$-0.477731\pi$
$593$	3.40453	0.139807	0.0699036	+	0.997554i	$0.477731\pi$
$594$	0	0
$595$	−34.2572	−1.40441
$596$	0	0
$597$	53.1109	2.17368
$598$	0	0
$599$	−12.9033	−0.527215	−0.263608	−	0.964630i	$-0.584912\pi$
$599$	−12.9033	−0.527215	−0.263608	+	0.964630i	$0.584912\pi$
$600$	0	0
$601$	−42.2865	−1.72490	−0.862451	−	0.506141i	$-0.831072\pi$
$601$	−42.2865	−1.72490	−0.862451	+	0.506141i	$0.831072\pi$
$602$	0	0
$603$	95.1959	3.87668
$604$	0	0
$605$	44.7773	1.82046
$606$	0	0
$607$	−26.5902	−1.07926	−0.539632	−	0.841901i	$-0.681437\pi$
$607$	−26.5902	−1.07926	−0.539632	+	0.841901i	$0.681437\pi$
$608$	0	0
$609$	34.8908	1.41385
$610$	0	0
$611$	−3.53845	−0.143150
$612$	0	0
$613$	38.0245	1.53580	0.767898	−	0.640572i	$-0.221303\pi$
$613$	38.0245	1.53580	0.767898	+	0.640572i	$0.221303\pi$
$614$	0	0
$615$	−127.028	−5.12228
$616$	0	0
$617$	−1.78959	−0.0720462	−0.0360231	−	0.999351i	$-0.511469\pi$
$617$	−1.78959	−0.0720462	−0.0360231	+	0.999351i	$0.511469\pi$
$618$	0	0
$619$	−28.1856	−1.13287	−0.566437	−	0.824105i	$-0.691679\pi$
$619$	−28.1856	−1.13287	−0.566437	+	0.824105i	$0.691679\pi$
$620$	0	0
$621$	23.3398	0.936594
$622$	0	0
$623$	−9.94481	−0.398430
$624$	0	0
$625$	59.1145	2.36458
$626$	0	0
$627$	−3.87030	−0.154565
$628$	0	0
$629$	−7.96395	−0.317543
$630$	0	0
$631$	22.2671	0.886440	0.443220	−	0.896413i	$-0.353836\pi$
$631$	22.2671	0.886440	0.443220	+	0.896413i	$0.353836\pi$
$632$	0	0
$633$	4.95916	0.197109
$634$	0	0
$635$	−37.2074	−1.47653
$636$	0	0
$637$	−3.22840	−0.127914
$638$	0	0
$639$	−99.4987	−3.93611
$640$	0	0
$641$	−13.2247	−0.522342	−0.261171	−	0.965292i	$-0.584109\pi$
$641$	−13.2247	−0.522342	−0.261171	+	0.965292i	$0.584109\pi$
$642$	0	0
$643$	−39.2888	−1.54940	−0.774700	−	0.632329i	$-0.782099\pi$
$643$	−39.2888	−1.54940	−0.774700	+	0.632329i	$0.782099\pi$
$644$	0	0
$645$	31.6834	1.24753
$646$	0	0
$647$	31.0960	1.22251	0.611256	−	0.791433i	$-0.290665\pi$
$647$	31.0960	1.22251	0.611256	+	0.791433i	$0.290665\pi$
$648$	0	0
$649$	−2.60247	−0.102156
$650$	0	0
$651$	8.37905	0.328401
$652$	0	0
$653$	−11.2439	−0.440009	−0.220005	−	0.975499i	$-0.570607\pi$
$653$	−11.2439	−0.440009	−0.220005	+	0.975499i	$0.570607\pi$
$654$	0	0
$655$	−10.3224	−0.403331
$656$	0	0
$657$	−73.2707	−2.85856
$658$	0	0
$659$	−38.5668	−1.50235	−0.751174	−	0.660104i	$-0.770512\pi$
$659$	−38.5668	−1.50235	−0.751174	+	0.660104i	$0.770512\pi$
$660$	0	0
$661$	9.47024	0.368350	0.184175	−	0.982894i	$-0.441039\pi$
$661$	9.47024	0.368350	0.184175	+	0.982894i	$0.441039\pi$
$662$	0	0
$663$	12.5875	0.488859
$664$	0	0
$665$	17.2432	0.668661
$666$	0	0
$667$	−12.3732	−0.479093
$668$	0	0
$669$	−5.49495	−0.212447
$670$	0	0
$671$	−1.46009	−0.0563660
$672$	0	0
$673$	−20.7639	−0.800390	−0.400195	−	0.916430i	$-0.631058\pi$
$673$	−20.7639	−0.800390	−0.400195	+	0.916430i	$0.631058\pi$
$674$	0	0
$675$	−182.314	−7.01728
$676$	0	0
$677$	−6.09922	−0.234412	−0.117206	−	0.993108i	$-0.537394\pi$
$677$	−6.09922	−0.234412	−0.117206	+	0.993108i	$0.537394\pi$
$678$	0	0
$679$	12.8535	0.493271
$680$	0	0
$681$	−91.9367	−3.52302
$682$	0	0
$683$	30.4890	1.16663	0.583315	−	0.812246i	$-0.301755\pi$
$683$	30.4890	1.16663	0.583315	+	0.812246i	$0.301755\pi$
$684$	0	0
$685$	−6.58759	−0.251699
$686$	0	0
$687$	26.1598	0.998059
$688$	0	0
$689$	−6.49813	−0.247559
$690$	0	0
$691$	45.9131	1.74662	0.873308	−	0.487169i	$-0.161970\pi$
$691$	45.9131	1.74662	0.873308	+	0.487169i	$0.161970\pi$
$692$	0	0
$693$	−3.82704	−0.145377
$694$	0	0
$695$	−29.5686	−1.12160
$696$	0	0
$697$	59.0399	2.23630
$698$	0	0
$699$	72.0219	2.72412
$700$	0	0
$701$	−40.4031	−1.52600	−0.763002	−	0.646396i	$-0.776275\pi$
$701$	−40.4031	−1.52600	−0.763002	+	0.646396i	$0.776275\pi$
$702$	0	0
$703$	4.00861	0.151188
$704$	0	0
$705$	−77.2508	−2.90943
$706$	0	0
$707$	−16.8651	−0.634277
$708$	0	0
$709$	33.9016	1.27320	0.636601	−	0.771193i	$-0.280340\pi$
$709$	33.9016	1.27320	0.636601	+	0.771193i	$0.280340\pi$
$710$	0	0
$711$	29.2587	1.09729
$712$	0	0
$713$	−2.97144	−0.111281
$714$	0	0
$715$	−0.957456	−0.0358068
$716$	0	0
$717$	−51.6640	−1.92943
$718$	0	0
$719$	−3.82323	−0.142582	−0.0712911	−	0.997456i	$-0.522712\pi$
$719$	−3.82323	−0.142582	−0.0712911	+	0.997456i	$0.522712\pi$
$720$	0	0
$721$	−4.70398	−0.175185
$722$	0	0
$723$	−68.2687	−2.53894
$724$	0	0
$725$	96.6511	3.58953
$726$	0	0
$727$	20.1793	0.748407	0.374204	−	0.927347i	$-0.377916\pi$
$727$	20.1793	0.748407	0.374204	+	0.927347i	$0.377916\pi$
$728$	0	0
$729$	54.9052	2.03353
$730$	0	0
$731$	−14.7257	−0.544651
$732$	0	0
$733$	−38.8068	−1.43336	−0.716680	−	0.697402i	$-0.754339\pi$
$733$	−38.8068	−1.43336	−0.716680	+	0.697402i	$0.754339\pi$
$734$	0	0
$735$	−70.4819	−2.59976
$736$	0	0
$737$	4.68477	0.172566
$738$	0	0
$739$	31.2874	1.15093	0.575463	−	0.817827i	$-0.304822\pi$
$739$	31.2874	1.15093	0.575463	+	0.817827i	$0.304822\pi$
$740$	0	0
$741$	−6.33586	−0.232754
$742$	0	0
$743$	−7.88088	−0.289121	−0.144561	−	0.989496i	$-0.546177\pi$
$743$	−7.88088	−0.289121	−0.144561	+	0.989496i	$0.546177\pi$
$744$	0	0
$745$	−18.7012	−0.685159
$746$	0	0
$747$	−56.6143	−2.07141
$748$	0	0
$749$	2.93271	0.107159
$750$	0	0
$751$	46.4043	1.69332	0.846659	−	0.532136i	$-0.178610\pi$
$751$	46.4043	1.69332	0.846659	+	0.532136i	$0.178610\pi$
$752$	0	0
$753$	11.9596	0.435831
$754$	0	0
$755$	64.8436	2.35990
$756$	0	0
$757$	20.1542	0.732516	0.366258	−	0.930513i	$-0.380639\pi$
$757$	20.1542	0.732516	0.366258	+	0.930513i	$0.380639\pi$
$758$	0	0
$759$	1.88922	0.0685742
$760$	0	0
$761$	−5.42345	−0.196600	−0.0983000	−	0.995157i	$-0.531340\pi$
$761$	−5.42345	−0.196600	−0.0983000	+	0.995157i	$0.531340\pi$
$762$	0	0
$763$	2.48981	0.0901373
$764$	0	0
$765$	197.416	7.13760
$766$	0	0
$767$	−4.26036	−0.153833
$768$	0	0
$769$	−16.1049	−0.580756	−0.290378	−	0.956912i	$-0.593781\pi$
$769$	−16.1049	−0.580756	−0.290378	+	0.956912i	$0.593781\pi$
$770$	0	0
$771$	8.08056	0.291014
$772$	0	0
$773$	−41.6700	−1.49877	−0.749383	−	0.662137i	$-0.769650\pi$
$773$	−41.6700	−1.49877	−0.749383	+	0.662137i	$0.769650\pi$
$774$	0	0
$775$	23.2108	0.833757
$776$	0	0
$777$	5.51771	0.197947
$778$	0	0
$779$	−29.7174	−1.06474
$780$	0	0
$781$	−4.89653	−0.175211
$782$	0	0
$783$	−122.244	−4.36866
$784$	0	0
$785$	6.50808	0.232283
$786$	0	0
$787$	2.92738	0.104350	0.0521748	−	0.998638i	$-0.483385\pi$
$787$	2.92738	0.104350	0.0521748	+	0.998638i	$0.483385\pi$
$788$	0	0
$789$	−26.8266	−0.955053
$790$	0	0
$791$	24.3170	0.864612
$792$	0	0
$793$	−2.39023	−0.0848795
$794$	0	0
$795$	−141.866	−5.03147
$796$	0	0
$797$	−6.95972	−0.246526	−0.123263	−	0.992374i	$-0.539336\pi$
$797$	−6.95972	−0.246526	−0.123263	+	0.992374i	$0.539336\pi$
$798$	0	0
$799$	35.9044	1.27021
$800$	0	0
$801$	57.3097	2.02494
$802$	0	0
$803$	−3.60579	−0.127246
$804$	0	0
$805$	−8.41692	−0.296658
$806$	0	0
$807$	2.02367	0.0712364
$808$	0	0
$809$	−28.4140	−0.998984	−0.499492	−	0.866319i	$-0.666480\pi$
$809$	−28.4140	−0.998984	−0.499492	+	0.866319i	$0.666480\pi$
$810$	0	0
$811$	−26.6184	−0.934697	−0.467348	−	0.884073i	$-0.654791\pi$
$811$	−26.6184	−0.934697	−0.467348	+	0.884073i	$0.654791\pi$
$812$	0	0
$813$	18.4950	0.648650
$814$	0	0
$815$	81.3268	2.84875
$816$	0	0
$817$	7.41212	0.259317
$818$	0	0
$819$	−6.26503	−0.218918
$820$	0	0
$821$	6.35923	0.221939	0.110969	−	0.993824i	$-0.464604\pi$
$821$	6.35923	0.221939	0.110969	+	0.993824i	$0.464604\pi$
$822$	0	0
$823$	−49.4153	−1.72251	−0.861255	−	0.508173i	$-0.830321\pi$
$823$	−49.4153	−1.72251	−0.861255	+	0.508173i	$0.830321\pi$
$824$	0	0
$825$	−14.7572	−0.513781
$826$	0	0
$827$	−27.2131	−0.946294	−0.473147	−	0.880984i	$-0.656882\pi$
$827$	−27.2131	−0.946294	−0.473147	+	0.880984i	$0.656882\pi$
$828$	0	0
$829$	15.4097	0.535201	0.267600	−	0.963530i	$-0.413769\pi$
$829$	15.4097	0.535201	0.267600	+	0.963530i	$0.413769\pi$
$830$	0	0
$831$	35.4666	1.23032
$832$	0	0
$833$	32.7584	1.13501
$834$	0	0
$835$	56.6167	1.95930
$836$	0	0
$837$	−29.3570	−1.01473
$838$	0	0
$839$	−20.1762	−0.696560	−0.348280	−	0.937391i	$-0.613234\pi$
$839$	−20.1762	−0.696560	−0.348280	+	0.937391i	$0.613234\pi$
$840$	0	0
$841$	35.8059	1.23469
$842$	0	0
$843$	59.3138	2.04287
$844$	0	0
$845$	52.0425	1.79032
$846$	0	0
$847$	14.4189	0.495441
$848$	0	0
$849$	−75.0898	−2.57708
$850$	0	0
$851$	−1.95673	−0.0670757
$852$	0	0
$853$	56.0677	1.91972	0.959861	−	0.280475i	$-0.0904920\pi$
$853$	56.0677	1.91972	0.959861	+	0.280475i	$0.0904920\pi$
$854$	0	0
$855$	−99.3684	−3.39833
$856$	0	0
$857$	−13.7529	−0.469790	−0.234895	−	0.972021i	$-0.575475\pi$
$857$	−13.7529	−0.469790	−0.234895	+	0.972021i	$0.575475\pi$
$858$	0	0
$859$	0.00909718	0.000310392 0	0.000155196	−	1.00000i	$-0.499951\pi$
$859$	0.00909718	0.000310392 0	0.000155196		1.00000i	$0.499951\pi$
$860$	0	0
$861$	−40.9050	−1.39404
$862$	0	0
$863$	−12.3244	−0.419528	−0.209764	−	0.977752i	$-0.567270\pi$
$863$	−12.3244	−0.419528	−0.209764	+	0.977752i	$0.567270\pi$
$864$	0	0
$865$	−10.1274	−0.344341
$866$	0	0
$867$	−72.2396	−2.45339
$868$	0	0
$869$	1.43988	0.0488445
$870$	0	0
$871$	7.66918	0.259860
$872$	0	0
$873$	−74.0716	−2.50694
$874$	0	0
$875$	38.3663	1.29702
$876$	0	0
$877$	14.4318	0.487327	0.243664	−	0.969860i	$-0.421651\pi$
$877$	14.4318	0.487327	0.243664	+	0.969860i	$0.421651\pi$
$878$	0	0
$879$	−35.0539	−1.18234
$880$	0	0
$881$	45.6687	1.53862	0.769309	−	0.638877i	$-0.220601\pi$
$881$	45.6687	1.53862	0.769309	+	0.638877i	$0.220601\pi$
$882$	0	0
$883$	20.7700	0.698967	0.349483	−	0.936943i	$-0.386357\pi$
$883$	20.7700	0.698967	0.349483	+	0.936943i	$0.386357\pi$
$884$	0	0
$885$	−93.0114	−3.12654
$886$	0	0
$887$	39.5552	1.32813	0.664066	−	0.747674i	$-0.268829\pi$
$887$	39.5552	1.32813	0.664066	+	0.747674i	$0.268829\pi$
$888$	0	0
$889$	−11.9813	−0.401841
$890$	0	0
$891$	10.0191	0.335653
$892$	0	0
$893$	−18.0723	−0.604766
$894$	0	0
$895$	−97.1892	−3.24868
$896$	0	0
$897$	3.09273	0.103263
$898$	0	0
$899$	15.5632	0.519061
$900$	0	0
$901$	65.9361	2.19665
$902$	0	0
$903$	10.2025	0.339518
$904$	0	0
$905$	38.1215	1.26720
$906$	0	0
$907$	16.2418	0.539300	0.269650	−	0.962958i	$-0.413092\pi$
$907$	16.2418	0.539300	0.269650	+	0.962958i	$0.413092\pi$
$908$	0	0
$909$	97.1897	3.22358
$910$	0	0
$911$	18.7151	0.620060	0.310030	−	0.950727i	$-0.399661\pi$
$911$	18.7151	0.620060	0.310030	+	0.950727i	$0.399661\pi$
$912$	0	0
$913$	−2.78610	−0.0922065
$914$	0	0
$915$	−52.1830	−1.72512
$916$	0	0
$917$	−3.32397	−0.109767
$918$	0	0
$919$	42.0824	1.38817	0.694085	−	0.719893i	$-0.255809\pi$
$919$	42.0824	1.38817	0.694085	+	0.719893i	$0.255809\pi$
$920$	0	0
$921$	28.0062	0.922835
$922$	0	0
$923$	−8.01583	−0.263844
$924$	0	0
$925$	15.2846	0.502554
$926$	0	0
$927$	27.1080	0.890343
$928$	0	0
$929$	15.2476	0.500257	0.250128	−	0.968213i	$-0.419527\pi$
$929$	15.2476	0.500257	0.250128	+	0.968213i	$0.419527\pi$
$930$	0	0
$931$	−16.4888	−0.540397
$932$	0	0
$933$	−94.8223	−3.10434
$934$	0	0
$935$	9.71524	0.317722
$936$	0	0
$937$	−33.0433	−1.07948	−0.539740	−	0.841832i	$-0.681477\pi$
$937$	−33.0433	−1.07948	−0.539740	+	0.841832i	$0.681477\pi$
$938$	0	0
$939$	38.2981	1.24981
$940$	0	0
$941$	31.7256	1.03423	0.517113	−	0.855917i	$-0.327007\pi$
$941$	31.7256	1.03423	0.517113	+	0.855917i	$0.327007\pi$
$942$	0	0
$943$	14.5060	0.472380
$944$	0	0
$945$	−83.1571	−2.70510
$946$	0	0
$947$	−31.5717	−1.02594	−0.512971	−	0.858406i	$-0.671455\pi$
$947$	−31.5717	−1.02594	−0.512971	+	0.858406i	$0.671455\pi$
$948$	0	0
$949$	−5.90284	−0.191614
$950$	0	0
$951$	−30.4003	−0.985799
$952$	0	0
$953$	5.40193	0.174986	0.0874929	−	0.996165i	$-0.472114\pi$
$953$	5.40193	0.174986	0.0874929	+	0.996165i	$0.472114\pi$
$954$	0	0
$955$	13.2740	0.429537
$956$	0	0
$957$	−9.89494	−0.319858
$958$	0	0
$959$	−2.12130	−0.0685002
$960$	0	0
$961$	−27.2625	−0.879435
$962$	0	0
$963$	−16.9005	−0.544612
$964$	0	0
$965$	−71.5390	−2.30292
$966$	0	0
$967$	−7.39238	−0.237723	−0.118861	−	0.992911i	$-0.537924\pi$
$967$	−7.39238	−0.237723	−0.118861	+	0.992911i	$0.537924\pi$
$968$	0	0
$969$	64.2895	2.06528
$970$	0	0
$971$	15.8777	0.509541	0.254770	−	0.967002i	$-0.418000\pi$
$971$	15.8777	0.509541	0.254770	+	0.967002i	$0.418000\pi$
$972$	0	0
$973$	−9.52150	−0.305245
$974$	0	0
$975$	−24.1583	−0.773684
$976$	0	0
$977$	5.30773	0.169809	0.0849047	−	0.996389i	$-0.472941\pi$
$977$	5.30773	0.169809	0.0849047	+	0.996389i	$0.472941\pi$
$978$	0	0
$979$	2.82032	0.0901378
$980$	0	0
$981$	−14.3482	−0.458103
$982$	0	0
$983$	27.8639	0.888720	0.444360	−	0.895848i	$-0.353431\pi$
$983$	27.8639	0.888720	0.444360	+	0.895848i	$0.353431\pi$
$984$	0	0
$985$	91.0186	2.90009
$986$	0	0
$987$	−24.8758	−0.791807
$988$	0	0
$989$	−3.61809	−0.115048
$990$	0	0
$991$	20.5478	0.652723	0.326362	−	0.945245i	$-0.394177\pi$
$991$	20.5478	0.652723	0.326362	+	0.945245i	$0.394177\pi$
$992$	0	0
$993$	19.4129	0.616050
$994$	0	0
$995$	67.1054	2.12738
$996$	0	0
$997$	−50.8094	−1.60915	−0.804575	−	0.593851i	$-0.797607\pi$
$997$	−50.8094	−1.60915	−0.804575	+	0.593851i	$0.797607\pi$
$998$	0	0
$999$	−19.3320	−0.611637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.2	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.2	✓	63	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-3.26383 q^{3} -4.12384 q^{5} -1.32793 q^{7} +7.65258 q^{9} +O(q^{10})\)	\(q-3.26383 q^{3} -4.12384 q^{5} -1.32793 q^{7} +7.65258 q^{9} +0.376598 q^{11} +0.616508 q^{13} +13.4595 q^{15} -6.25567 q^{17} +3.14876 q^{19} +4.33415 q^{21} -1.53701 q^{23} +12.0060 q^{25} -15.1852 q^{27} +8.05021 q^{29} +1.93326 q^{31} -1.22915 q^{33} +5.47618 q^{35} +1.27308 q^{37} -2.01218 q^{39} -9.43783 q^{41} +2.35398 q^{43} -31.5580 q^{45} -5.73950 q^{47} -5.23659 q^{49} +20.4174 q^{51} -10.5402 q^{53} -1.55303 q^{55} -10.2770 q^{57} -6.91047 q^{59} -3.87704 q^{61} -10.1621 q^{63} -2.54238 q^{65} +12.4397 q^{67} +5.01653 q^{69} -13.0020 q^{71} -9.57463 q^{73} -39.1856 q^{75} -0.500098 q^{77} +3.82338 q^{79} +26.6043 q^{81} -7.39807 q^{83} +25.7974 q^{85} -26.2745 q^{87} +7.48893 q^{89} -0.818682 q^{91} -6.30984 q^{93} -12.9850 q^{95} -9.67930 q^{97} +2.88195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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