LMFDB - Embedded newform 6044.2.a.b.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.68247 q^{3} +0.714038 q^{5} +3.91380 q^{7} +4.19563 q^{9} +O(q^{10})$	$q-2.68247 q^{3} +0.714038 q^{5} +3.91380 q^{7} +4.19563 q^{9} -4.61641 q^{11} -1.76023 q^{13} -1.91538 q^{15} -5.97903 q^{17} +0.502363 q^{19} -10.4986 q^{21} -5.20043 q^{23} -4.49015 q^{25} -3.20725 q^{27} -8.46239 q^{29} +0.243320 q^{31} +12.3834 q^{33} +2.79460 q^{35} +11.8450 q^{37} +4.72175 q^{39} -1.42594 q^{41} -1.16360 q^{43} +2.99584 q^{45} -2.46441 q^{47} +8.31780 q^{49} +16.0386 q^{51} +12.2982 q^{53} -3.29630 q^{55} -1.34757 q^{57} -0.914030 q^{59} +3.87284 q^{61} +16.4209 q^{63} -1.25687 q^{65} +13.5984 q^{67} +13.9500 q^{69} -1.15135 q^{71} -12.5294 q^{73} +12.0447 q^{75} -18.0677 q^{77} +4.26631 q^{79} -3.98356 q^{81} +7.60803 q^{83} -4.26926 q^{85} +22.7001 q^{87} -1.15607 q^{89} -6.88917 q^{91} -0.652698 q^{93} +0.358707 q^{95} +6.48371 q^{97} -19.3688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.68247	−1.54872	−0.774362	−	0.632743i	$-0.781929\pi$
$3$	−2.68247	−1.54872	−0.774362	+	0.632743i	$0.781929\pi$
$4$	0	0
$5$	0.714038	0.319328	0.159664	−	0.987171i	$-0.448959\pi$
$5$	0.714038	0.319328	0.159664	+	0.987171i	$0.448959\pi$
$6$	0	0
$7$	3.91380	1.47928	0.739638	−	0.673005i	$-0.234997\pi$
$7$	3.91380	1.47928	0.739638	+	0.673005i	$0.234997\pi$
$8$	0	0
$9$	4.19563	1.39854
$10$	0	0
$11$	−4.61641	−1.39190	−0.695950	−	0.718090i	$-0.745017\pi$
$11$	−4.61641	−1.39190	−0.695950	+	0.718090i	$0.745017\pi$
$12$	0	0
$13$	−1.76023	−0.488199	−0.244100	−	0.969750i	$-0.578492\pi$
$13$	−1.76023	−0.488199	−0.244100	+	0.969750i	$0.578492\pi$
$14$	0	0
$15$	−1.91538	−0.494550
$16$	0	0
$17$	−5.97903	−1.45013	−0.725064	−	0.688682i	$-0.758190\pi$
$17$	−5.97903	−1.45013	−0.725064	+	0.688682i	$0.758190\pi$
$18$	0	0
$19$	0.502363	0.115250	0.0576250	−	0.998338i	$-0.481647\pi$
$19$	0.502363	0.115250	0.0576250	+	0.998338i	$0.481647\pi$
$20$	0	0
$21$	−10.4986	−2.29099
$22$	0	0
$23$	−5.20043	−1.08436	−0.542182	−	0.840261i	$-0.682402\pi$
$23$	−5.20043	−1.08436	−0.542182	+	0.840261i	$0.682402\pi$
$24$	0	0
$25$	−4.49015	−0.898030
$26$	0	0
$27$	−3.20725	−0.617235
$28$	0	0
$29$	−8.46239	−1.57143	−0.785713	−	0.618591i	$-0.787704\pi$
$29$	−8.46239	−1.57143	−0.785713	+	0.618591i	$0.787704\pi$
$30$	0	0
$31$	0.243320	0.0437016	0.0218508	−	0.999761i	$-0.493044\pi$
$31$	0.243320	0.0437016	0.0218508	+	0.999761i	$0.493044\pi$
$32$	0	0
$33$	12.3834	2.15567
$34$	0	0
$35$	2.79460	0.472374
$36$	0	0
$37$	11.8450	1.94730	0.973651	−	0.228043i	$-0.0732328\pi$
$37$	11.8450	1.94730	0.973651	+	0.228043i	$0.0732328\pi$
$38$	0	0
$39$	4.72175	0.756086
$40$	0	0
$41$	−1.42594	−0.222695	−0.111347	−	0.993782i	$-0.535517\pi$
$41$	−1.42594	−0.222695	−0.111347	+	0.993782i	$0.535517\pi$
$42$	0	0
$43$	−1.16360	−0.177448	−0.0887238	−	0.996056i	$-0.528279\pi$
$43$	−1.16360	−0.177448	−0.0887238	+	0.996056i	$0.528279\pi$
$44$	0	0
$45$	2.99584	0.446594
$46$	0	0
$47$	−2.46441	−0.359471	−0.179736	−	0.983715i	$-0.557524\pi$
$47$	−2.46441	−0.359471	−0.179736	+	0.983715i	$0.557524\pi$
$48$	0	0
$49$	8.31780	1.18826
$50$	0	0
$51$	16.0386	2.24585
$52$	0	0
$53$	12.2982	1.68929	0.844643	−	0.535331i	$-0.179813\pi$
$53$	12.2982	1.68929	0.844643	+	0.535331i	$0.179813\pi$
$54$	0	0
$55$	−3.29630	−0.444472
$56$	0	0
$57$	−1.34757	−0.178490
$58$	0	0
$59$	−0.914030	−0.118997	−0.0594983	−	0.998228i	$-0.518950\pi$
$59$	−0.914030	−0.118997	−0.0594983	+	0.998228i	$0.518950\pi$
$60$	0	0
$61$	3.87284	0.495866	0.247933	−	0.968777i	$-0.420249\pi$
$61$	3.87284	0.495866	0.247933	+	0.968777i	$0.420249\pi$
$62$	0	0
$63$	16.4209	2.06883
$64$	0	0
$65$	−1.25687	−0.155896
$66$	0	0
$67$	13.5984	1.66131	0.830655	−	0.556788i	$-0.187966\pi$
$67$	13.5984	1.66131	0.830655	+	0.556788i	$0.187966\pi$
$68$	0	0
$69$	13.9500	1.67938
$70$	0	0
$71$	−1.15135	−0.136640	−0.0683199	−	0.997663i	$-0.521764\pi$
$71$	−1.15135	−0.136640	−0.0683199	+	0.997663i	$0.521764\pi$
$72$	0	0
$73$	−12.5294	−1.46646	−0.733228	−	0.679982i	$-0.761987\pi$
$73$	−12.5294	−1.46646	−0.733228	+	0.679982i	$0.761987\pi$
$74$	0	0
$75$	12.0447	1.39080
$76$	0	0
$77$	−18.0677	−2.05901
$78$	0	0
$79$	4.26631	0.479998	0.239999	−	0.970773i	$-0.422853\pi$
$79$	4.26631	0.479998	0.239999	+	0.970773i	$0.422853\pi$
$80$	0	0
$81$	−3.98356	−0.442618
$82$	0	0
$83$	7.60803	0.835089	0.417545	−	0.908656i	$-0.362891\pi$
$83$	7.60803	0.835089	0.417545	+	0.908656i	$0.362891\pi$
$84$	0	0
$85$	−4.26926	−0.463066
$86$	0	0
$87$	22.7001	2.43370
$88$	0	0
$89$	−1.15607	−0.122543	−0.0612715	−	0.998121i	$-0.519516\pi$
$89$	−1.15607	−0.122543	−0.0612715	+	0.998121i	$0.519516\pi$
$90$	0	0
$91$	−6.88917	−0.722181
$92$	0	0
$93$	−0.652698	−0.0676816
$94$	0	0
$95$	0.358707	0.0368025
$96$	0	0
$97$	6.48371	0.658321	0.329160	−	0.944274i	$-0.393234\pi$
$97$	6.48371	0.658321	0.329160	+	0.944274i	$0.393234\pi$
$98$	0	0
$99$	−19.3688	−1.94664
$100$	0	0
$101$	−3.58600	−0.356821	−0.178410	−	0.983956i	$-0.557095\pi$
$101$	−3.58600	−0.356821	−0.178410	+	0.983956i	$0.557095\pi$
$102$	0	0
$103$	13.8718	1.36682	0.683412	−	0.730033i	$-0.260495\pi$
$103$	13.8718	1.36682	0.683412	+	0.730033i	$0.260495\pi$
$104$	0	0
$105$	−7.49642	−0.731576
$106$	0	0
$107$	−6.39045	−0.617788	−0.308894	−	0.951097i	$-0.599959\pi$
$107$	−6.39045	−0.617788	−0.308894	+	0.951097i	$0.599959\pi$
$108$	0	0
$109$	4.27817	0.409774	0.204887	−	0.978786i	$-0.434317\pi$
$109$	4.27817	0.409774	0.204887	+	0.978786i	$0.434317\pi$
$110$	0	0
$111$	−31.7738	−3.01583
$112$	0	0
$113$	−16.1569	−1.51992	−0.759959	−	0.649971i	$-0.774781\pi$
$113$	−16.1569	−1.51992	−0.759959	+	0.649971i	$0.774781\pi$
$114$	0	0
$115$	−3.71330	−0.346267
$116$	0	0
$117$	−7.38527	−0.682768
$118$	0	0
$119$	−23.4007	−2.14514
$120$	0	0
$121$	10.3113	0.937388
$122$	0	0
$123$	3.82504	0.344893
$124$	0	0
$125$	−6.77633	−0.606093
$126$	0	0
$127$	13.8714	1.23089	0.615443	−	0.788181i	$-0.288977\pi$
$127$	13.8714	1.23089	0.615443	+	0.788181i	$0.288977\pi$
$128$	0	0
$129$	3.12132	0.274817
$130$	0	0
$131$	20.4952	1.79068	0.895338	−	0.445387i	$-0.146934\pi$
$131$	20.4952	1.79068	0.895338	+	0.445387i	$0.146934\pi$
$132$	0	0
$133$	1.96615	0.170487
$134$	0	0
$135$	−2.29010	−0.197100
$136$	0	0
$137$	6.81783	0.582487	0.291243	−	0.956649i	$-0.405931\pi$
$137$	6.81783	0.582487	0.291243	+	0.956649i	$0.405931\pi$
$138$	0	0
$139$	−0.448549	−0.0380454	−0.0190227	−	0.999819i	$-0.506055\pi$
$139$	−0.448549	−0.0380454	−0.0190227	+	0.999819i	$0.506055\pi$
$140$	0	0
$141$	6.61070	0.556722
$142$	0	0
$143$	8.12594	0.679525
$144$	0	0
$145$	−6.04247	−0.501800
$146$	0	0
$147$	−22.3122	−1.84028
$148$	0	0
$149$	7.71917	0.632379	0.316189	−	0.948696i	$-0.397597\pi$
$149$	7.71917	0.632379	0.316189	+	0.948696i	$0.397597\pi$
$150$	0	0
$151$	−19.7224	−1.60499	−0.802494	−	0.596661i	$-0.796494\pi$
$151$	−19.7224	−1.60499	−0.802494	+	0.596661i	$0.796494\pi$
$152$	0	0
$153$	−25.0858	−2.02807
$154$	0	0
$155$	0.173740	0.0139551
$156$	0	0
$157$	−19.0593	−1.52110	−0.760549	−	0.649280i	$-0.775070\pi$
$157$	−19.0593	−1.52110	−0.760549	+	0.649280i	$0.775070\pi$
$158$	0	0
$159$	−32.9895	−2.61624
$160$	0	0
$161$	−20.3534	−1.60407
$162$	0	0
$163$	5.47482	0.428821	0.214410	−	0.976744i	$-0.431217\pi$
$163$	5.47482	0.428821	0.214410	+	0.976744i	$0.431217\pi$
$164$	0	0
$165$	8.84221	0.688365
$166$	0	0
$167$	−13.5840	−1.05116	−0.525580	−	0.850744i	$-0.676152\pi$
$167$	−13.5840	−1.05116	−0.525580	+	0.850744i	$0.676152\pi$
$168$	0	0
$169$	−9.90160	−0.761661
$170$	0	0
$171$	2.10773	0.161182
$172$	0	0
$173$	23.9333	1.81962	0.909808	−	0.415030i	$-0.136229\pi$
$173$	23.9333	1.81962	0.909808	+	0.415030i	$0.136229\pi$
$174$	0	0
$175$	−17.5735	−1.32843
$176$	0	0
$177$	2.45186	0.184293
$178$	0	0
$179$	−0.541051	−0.0404400	−0.0202200	−	0.999796i	$-0.506437\pi$
$179$	−0.541051	−0.0404400	−0.0202200	+	0.999796i	$0.506437\pi$
$180$	0	0
$181$	24.8056	1.84379	0.921893	−	0.387445i	$-0.126642\pi$
$181$	24.8056	1.84379	0.921893	+	0.387445i	$0.126642\pi$
$182$	0	0
$183$	−10.3888	−0.767960
$184$	0	0
$185$	8.45777	0.621827
$186$	0	0
$187$	27.6017	2.01843
$188$	0	0
$189$	−12.5525	−0.913061
$190$	0	0
$191$	20.4841	1.48218	0.741090	−	0.671406i	$-0.234309\pi$
$191$	20.4841	1.48218	0.741090	+	0.671406i	$0.234309\pi$
$192$	0	0
$193$	5.87470	0.422870	0.211435	−	0.977392i	$-0.432186\pi$
$193$	5.87470	0.422870	0.211435	+	0.977392i	$0.432186\pi$
$194$	0	0
$195$	3.37151	0.241439
$196$	0	0
$197$	−18.9647	−1.35118	−0.675590	−	0.737277i	$-0.736111\pi$
$197$	−18.9647	−1.35118	−0.675590	+	0.737277i	$0.736111\pi$
$198$	0	0
$199$	7.53400	0.534071	0.267036	−	0.963687i	$-0.413956\pi$
$199$	7.53400	0.534071	0.267036	+	0.963687i	$0.413956\pi$
$200$	0	0
$201$	−36.4773	−2.57291
$202$	0	0
$203$	−33.1201	−2.32457
$204$	0	0
$205$	−1.01818	−0.0711126
$206$	0	0
$207$	−21.8191	−1.51653
$208$	0	0
$209$	−2.31912	−0.160417
$210$	0	0
$211$	22.6733	1.56090	0.780448	−	0.625221i	$-0.214991\pi$
$211$	22.6733	1.56090	0.780448	+	0.625221i	$0.214991\pi$
$212$	0	0
$213$	3.08845	0.211617
$214$	0	0
$215$	−0.830856	−0.0566639
$216$	0	0
$217$	0.952305	0.0646467
$218$	0	0
$219$	33.6097	2.27114
$220$	0	0
$221$	10.5245	0.707951
$222$	0	0
$223$	27.7240	1.85654	0.928268	−	0.371911i	$-0.121297\pi$
$223$	27.7240	1.85654	0.928268	+	0.371911i	$0.121297\pi$
$224$	0	0
$225$	−18.8390	−1.25593
$226$	0	0
$227$	2.78079	0.184568	0.0922838	−	0.995733i	$-0.470583\pi$
$227$	2.78079	0.184568	0.0922838	+	0.995733i	$0.470583\pi$
$228$	0	0
$229$	4.08530	0.269964	0.134982	−	0.990848i	$-0.456902\pi$
$229$	4.08530	0.269964	0.134982	+	0.990848i	$0.456902\pi$
$230$	0	0
$231$	48.4660	3.18883
$232$	0	0
$233$	20.7900	1.36200	0.680999	−	0.732284i	$-0.261546\pi$
$233$	20.7900	1.36200	0.680999	+	0.732284i	$0.261546\pi$
$234$	0	0
$235$	−1.75968	−0.114789
$236$	0	0
$237$	−11.4442	−0.743383
$238$	0	0
$239$	17.0978	1.10596	0.552981	−	0.833194i	$-0.313490\pi$
$239$	17.0978	1.10596	0.552981	+	0.833194i	$0.313490\pi$
$240$	0	0
$241$	−22.9640	−1.47924	−0.739622	−	0.673023i	$-0.764995\pi$
$241$	−22.9640	−1.47924	−0.739622	+	0.673023i	$0.764995\pi$
$242$	0	0
$243$	20.3075	1.30273
$244$	0	0
$245$	5.93923	0.379443
$246$	0	0
$247$	−0.884274	−0.0562650
$248$	0	0
$249$	−20.4083	−1.29332
$250$	0	0
$251$	−16.1407	−1.01879	−0.509394	−	0.860533i	$-0.670131\pi$
$251$	−16.1407	−1.01879	−0.509394	+	0.860533i	$0.670131\pi$
$252$	0	0
$253$	24.0073	1.50933
$254$	0	0
$255$	11.4521	0.717161
$256$	0	0
$257$	−5.76658	−0.359710	−0.179855	−	0.983693i	$-0.557563\pi$
$257$	−5.76658	−0.359710	−0.179855	+	0.983693i	$0.557563\pi$
$258$	0	0
$259$	46.3588	2.88060
$260$	0	0
$261$	−35.5051	−2.19771
$262$	0	0
$263$	4.61698	0.284695	0.142348	−	0.989817i	$-0.454535\pi$
$263$	4.61698	0.284695	0.142348	+	0.989817i	$0.454535\pi$
$264$	0	0
$265$	8.78137	0.539435
$266$	0	0
$267$	3.10112	0.189785
$268$	0	0
$269$	26.3085	1.60406	0.802028	−	0.597287i	$-0.203755\pi$
$269$	26.3085	1.60406	0.802028	+	0.597287i	$0.203755\pi$
$270$	0	0
$271$	−2.62910	−0.159706	−0.0798531	−	0.996807i	$-0.525445\pi$
$271$	−2.62910	−0.159706	−0.0798531	+	0.996807i	$0.525445\pi$
$272$	0	0
$273$	18.4800	1.11846
$274$	0	0
$275$	20.7284	1.24997
$276$	0	0
$277$	−27.3026	−1.64045	−0.820226	−	0.572040i	$-0.806152\pi$
$277$	−27.3026	−1.64045	−0.820226	+	0.572040i	$0.806152\pi$
$278$	0	0
$279$	1.02088	0.0611186
$280$	0	0
$281$	−27.2370	−1.62482	−0.812412	−	0.583084i	$-0.801846\pi$
$281$	−27.2370	−1.62482	−0.812412	+	0.583084i	$0.801846\pi$
$282$	0	0
$283$	−23.0489	−1.37011	−0.685056	−	0.728490i	$-0.740222\pi$
$283$	−23.0489	−1.37011	−0.685056	+	0.728490i	$0.740222\pi$
$284$	0	0
$285$	−0.962219	−0.0569969
$286$	0	0
$287$	−5.58085	−0.329427
$288$	0	0
$289$	18.7488	1.10287
$290$	0	0
$291$	−17.3923	−1.01956
$292$	0	0
$293$	−11.0567	−0.645937	−0.322968	−	0.946410i	$-0.604681\pi$
$293$	−11.0567	−0.645937	−0.322968	+	0.946410i	$0.604681\pi$
$294$	0	0
$295$	−0.652653	−0.0379989
$296$	0	0
$297$	14.8060	0.859130
$298$	0	0
$299$	9.15394	0.529386
$300$	0	0
$301$	−4.55410	−0.262494
$302$	0	0
$303$	9.61933	0.552616
$304$	0	0
$305$	2.76536	0.158344
$306$	0	0
$307$	9.53595	0.544245	0.272123	−	0.962263i	$-0.412274\pi$
$307$	9.53595	0.544245	0.272123	+	0.962263i	$0.412274\pi$
$308$	0	0
$309$	−37.2105	−2.11683
$310$	0	0
$311$	33.1064	1.87729	0.938646	−	0.344881i	$-0.112081\pi$
$311$	33.1064	1.87729	0.938646	+	0.344881i	$0.112081\pi$
$312$	0	0
$313$	20.3331	1.14929	0.574647	−	0.818401i	$-0.305139\pi$
$313$	20.3331	1.14929	0.574647	+	0.818401i	$0.305139\pi$
$314$	0	0
$315$	11.7251	0.660635
$316$	0	0
$317$	−25.8764	−1.45336	−0.726682	−	0.686974i	$-0.758939\pi$
$317$	−25.8764	−1.45336	−0.726682	+	0.686974i	$0.758939\pi$
$318$	0	0
$319$	39.0659	2.18727
$320$	0	0
$321$	17.1422	0.956782
$322$	0	0
$323$	−3.00365	−0.167127
$324$	0	0
$325$	7.90369	0.438418
$326$	0	0
$327$	−11.4761	−0.634627
$328$	0	0
$329$	−9.64520	−0.531757
$330$	0	0
$331$	24.2211	1.33131	0.665657	−	0.746258i	$-0.268152\pi$
$331$	24.2211	1.33131	0.665657	+	0.746258i	$0.268152\pi$
$332$	0	0
$333$	49.6972	2.72339
$334$	0	0
$335$	9.70978	0.530502
$336$	0	0
$337$	19.9697	1.08782	0.543911	−	0.839143i	$-0.316943\pi$
$337$	19.9697	1.08782	0.543911	+	0.839143i	$0.316943\pi$
$338$	0	0
$339$	43.3405	2.35393
$340$	0	0
$341$	−1.12327	−0.0608282
$342$	0	0
$343$	5.15759	0.278484
$344$	0	0
$345$	9.96082	0.536272
$346$	0	0
$347$	2.61666	0.140469	0.0702347	−	0.997530i	$-0.477625\pi$
$347$	2.61666	0.140469	0.0702347	+	0.997530i	$0.477625\pi$
$348$	0	0
$349$	−23.9125	−1.28001	−0.640003	−	0.768373i	$-0.721067\pi$
$349$	−23.9125	−1.28001	−0.640003	+	0.768373i	$0.721067\pi$
$350$	0	0
$351$	5.64549	0.301334
$352$	0	0
$353$	−26.7046	−1.42134	−0.710670	−	0.703525i	$-0.751608\pi$
$353$	−26.7046	−1.42134	−0.710670	+	0.703525i	$0.751608\pi$
$354$	0	0
$355$	−0.822106	−0.0436329
$356$	0	0
$357$	62.7716	3.32223
$358$	0	0
$359$	4.60806	0.243204	0.121602	−	0.992579i	$-0.461197\pi$
$359$	4.60806	0.243204	0.121602	+	0.992579i	$0.461197\pi$
$360$	0	0
$361$	−18.7476	−0.986717
$362$	0	0
$363$	−27.6596	−1.45175
$364$	0	0
$365$	−8.94648	−0.468280
$366$	0	0
$367$	−30.6717	−1.60105	−0.800525	−	0.599299i	$-0.795446\pi$
$367$	−30.6717	−1.60105	−0.800525	+	0.599299i	$0.795446\pi$
$368$	0	0
$369$	−5.98273	−0.311449
$370$	0	0
$371$	48.1326	2.49892
$372$	0	0
$373$	24.6649	1.27710	0.638551	−	0.769580i	$-0.279534\pi$
$373$	24.6649	1.27710	0.638551	+	0.769580i	$0.279534\pi$
$374$	0	0
$375$	18.1773	0.938671
$376$	0	0
$377$	14.8957	0.767169
$378$	0	0
$379$	−0.0884132	−0.00454148	−0.00227074	−	0.999997i	$-0.500723\pi$
$379$	−0.0884132	−0.00454148	−0.00227074	+	0.999997i	$0.500723\pi$
$380$	0	0
$381$	−37.2095	−1.90630
$382$	0	0
$383$	−12.6577	−0.646778	−0.323389	−	0.946266i	$-0.604822\pi$
$383$	−12.6577	−0.646778	−0.323389	+	0.946266i	$0.604822\pi$
$384$	0	0
$385$	−12.9010	−0.657497
$386$	0	0
$387$	−4.88205	−0.248168
$388$	0	0
$389$	18.7750	0.951930	0.475965	−	0.879464i	$-0.342099\pi$
$389$	18.7750	0.951930	0.475965	+	0.879464i	$0.342099\pi$
$390$	0	0
$391$	31.0935	1.57247
$392$	0	0
$393$	−54.9778	−2.77326
$394$	0	0
$395$	3.04631	0.153276
$396$	0	0
$397$	−21.5129	−1.07970	−0.539850	−	0.841761i	$-0.681519\pi$
$397$	−21.5129	−1.07970	−0.539850	+	0.841761i	$0.681519\pi$
$398$	0	0
$399$	−5.27413	−0.264037
$400$	0	0
$401$	−23.0489	−1.15101	−0.575505	−	0.817798i	$-0.695194\pi$
$401$	−23.0489	−1.15101	−0.575505	+	0.817798i	$0.695194\pi$
$402$	0	0
$403$	−0.428299	−0.0213351
$404$	0	0
$405$	−2.84442	−0.141340
$406$	0	0
$407$	−54.6813	−2.71045
$408$	0	0
$409$	−20.9106	−1.03396	−0.516981	−	0.855997i	$-0.672944\pi$
$409$	−20.9106	−1.03396	−0.516981	+	0.855997i	$0.672944\pi$
$410$	0	0
$411$	−18.2886	−0.902111
$412$	0	0
$413$	−3.57733	−0.176029
$414$	0	0
$415$	5.43242	0.266667
$416$	0	0
$417$	1.20322	0.0589218
$418$	0	0
$419$	12.8007	0.625355	0.312677	−	0.949859i	$-0.398774\pi$
$419$	12.8007	0.625355	0.312677	+	0.949859i	$0.398774\pi$
$420$	0	0
$421$	36.5767	1.78264	0.891320	−	0.453375i	$-0.149780\pi$
$421$	36.5767	1.78264	0.891320	+	0.453375i	$0.149780\pi$
$422$	0	0
$423$	−10.3398	−0.502736
$424$	0	0
$425$	26.8467	1.30226
$426$	0	0
$427$	15.1575	0.733523
$428$	0	0
$429$	−21.7976	−1.05240
$430$	0	0
$431$	14.7750	0.711686	0.355843	−	0.934546i	$-0.384194\pi$
$431$	14.7750	0.711686	0.355843	+	0.934546i	$0.384194\pi$
$432$	0	0
$433$	12.0229	0.577784	0.288892	−	0.957362i	$-0.406713\pi$
$433$	12.0229	0.577784	0.288892	+	0.957362i	$0.406713\pi$
$434$	0	0
$435$	16.2087	0.777149
$436$	0	0
$437$	−2.61250	−0.124973
$438$	0	0
$439$	−9.97814	−0.476230	−0.238115	−	0.971237i	$-0.576530\pi$
$439$	−9.97814	−0.476230	−0.238115	+	0.971237i	$0.576530\pi$
$440$	0	0
$441$	34.8984	1.66183
$442$	0	0
$443$	30.1562	1.43276	0.716382	−	0.697709i	$-0.245797\pi$
$443$	30.1562	1.43276	0.716382	+	0.697709i	$0.245797\pi$
$444$	0	0
$445$	−0.825477	−0.0391314
$446$	0	0
$447$	−20.7064	−0.979380
$448$	0	0
$449$	23.8150	1.12390	0.561949	−	0.827172i	$-0.310052\pi$
$449$	23.8150	1.12390	0.561949	+	0.827172i	$0.310052\pi$
$450$	0	0
$451$	6.58274	0.309969
$452$	0	0
$453$	52.9047	2.48568
$454$	0	0
$455$	−4.91913	−0.230612
$456$	0	0
$457$	−0.596274	−0.0278925	−0.0139463	−	0.999903i	$-0.504439\pi$
$457$	−0.596274	−0.0278925	−0.0139463	+	0.999903i	$0.504439\pi$
$458$	0	0
$459$	19.1762	0.895070
$460$	0	0
$461$	31.1159	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$461$	31.1159	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$462$	0	0
$463$	−27.9585	−1.29934	−0.649670	−	0.760216i	$-0.725093\pi$
$463$	−27.9585	−1.29934	−0.649670	+	0.760216i	$0.725093\pi$
$464$	0	0
$465$	−0.466051	−0.0216126
$466$	0	0
$467$	33.3664	1.54401	0.772006	−	0.635615i	$-0.219253\pi$
$467$	33.3664	1.54401	0.772006	+	0.635615i	$0.219253\pi$
$468$	0	0
$469$	53.2214	2.45753
$470$	0	0
$471$	51.1260	2.35576
$472$	0	0
$473$	5.37167	0.246990
$474$	0	0
$475$	−2.25569	−0.103498
$476$	0	0
$477$	51.5987	2.36254
$478$	0	0
$479$	−5.43909	−0.248518	−0.124259	−	0.992250i	$-0.539655\pi$
$479$	−5.43909	−0.248518	−0.124259	+	0.992250i	$0.539655\pi$
$480$	0	0
$481$	−20.8499	−0.950672
$482$	0	0
$483$	54.5974	2.48427
$484$	0	0
$485$	4.62961	0.210220
$486$	0	0
$487$	16.6344	0.753779	0.376889	−	0.926258i	$-0.376994\pi$
$487$	16.6344	0.753779	0.376889	+	0.926258i	$0.376994\pi$
$488$	0	0
$489$	−14.6860	−0.664125
$490$	0	0
$491$	−0.0683538	−0.00308477	−0.00154238	−	0.999999i	$-0.500491\pi$
$491$	−0.0683538	−0.00308477	−0.00154238	+	0.999999i	$0.500491\pi$
$492$	0	0
$493$	50.5969	2.27877
$494$	0	0
$495$	−13.8300	−0.621614
$496$	0	0
$497$	−4.50614	−0.202128
$498$	0	0
$499$	−12.9589	−0.580121	−0.290061	−	0.957008i	$-0.593675\pi$
$499$	−12.9589	−0.580121	−0.290061	+	0.957008i	$0.593675\pi$
$500$	0	0
$501$	36.4386	1.62796
$502$	0	0
$503$	30.8319	1.37473	0.687364	−	0.726313i	$-0.258768\pi$
$503$	30.8319	1.37473	0.687364	+	0.726313i	$0.258768\pi$
$504$	0	0
$505$	−2.56054	−0.113943
$506$	0	0
$507$	26.5607	1.17960
$508$	0	0
$509$	−31.7634	−1.40789	−0.703943	−	0.710256i	$-0.748579\pi$
$509$	−31.7634	−1.40789	−0.703943	+	0.710256i	$0.748579\pi$
$510$	0	0
$511$	−49.0376	−2.16929
$512$	0	0
$513$	−1.61120	−0.0711364
$514$	0	0
$515$	9.90496	0.436465
$516$	0	0
$517$	11.3767	0.500348
$518$	0	0
$519$	−64.2003	−2.81808
$520$	0	0
$521$	20.9919	0.919670	0.459835	−	0.888004i	$-0.347908\pi$
$521$	20.9919	0.919670	0.459835	+	0.888004i	$0.347908\pi$
$522$	0	0
$523$	15.6615	0.684831	0.342415	−	0.939549i	$-0.388755\pi$
$523$	15.6615	0.684831	0.342415	+	0.939549i	$0.388755\pi$
$524$	0	0
$525$	47.1404	2.05738
$526$	0	0
$527$	−1.45482	−0.0633728
$528$	0	0
$529$	4.04445	0.175846
$530$	0	0
$531$	−3.83494	−0.166422
$532$	0	0
$533$	2.50998	0.108719
$534$	0	0
$535$	−4.56302	−0.197277
$536$	0	0
$537$	1.45135	0.0626304
$538$	0	0
$539$	−38.3984	−1.65394
$540$	0	0
$541$	−15.7483	−0.677074	−0.338537	−	0.940953i	$-0.609932\pi$
$541$	−15.7483	−0.677074	−0.338537	+	0.940953i	$0.609932\pi$
$542$	0	0
$543$	−66.5402	−2.85551
$544$	0	0
$545$	3.05478	0.130852
$546$	0	0
$547$	12.6499	0.540869	0.270435	−	0.962738i	$-0.412833\pi$
$547$	12.6499	0.540869	0.270435	+	0.962738i	$0.412833\pi$
$548$	0	0
$549$	16.2490	0.693491
$550$	0	0
$551$	−4.25119	−0.181107
$552$	0	0
$553$	16.6975	0.710049
$554$	0	0
$555$	−22.6877	−0.963038
$556$	0	0
$557$	−19.5622	−0.828879	−0.414439	−	0.910077i	$-0.636022\pi$
$557$	−19.5622	−0.828879	−0.414439	+	0.910077i	$0.636022\pi$
$558$	0	0
$559$	2.04820	0.0866298
$560$	0	0
$561$	−74.0406	−3.12600
$562$	0	0
$563$	37.8946	1.59706	0.798532	−	0.601952i	$-0.205610\pi$
$563$	37.8946	1.59706	0.798532	+	0.601952i	$0.205610\pi$
$564$	0	0
$565$	−11.5367	−0.485352
$566$	0	0
$567$	−15.5908	−0.654754
$568$	0	0
$569$	17.1106	0.717314	0.358657	−	0.933469i	$-0.383235\pi$
$569$	17.1106	0.717314	0.358657	+	0.933469i	$0.383235\pi$
$570$	0	0
$571$	29.1748	1.22093	0.610464	−	0.792044i	$-0.290983\pi$
$571$	29.1748	1.22093	0.610464	+	0.792044i	$0.290983\pi$
$572$	0	0
$573$	−54.9480	−2.29549
$574$	0	0
$575$	23.3507	0.973791
$576$	0	0
$577$	26.3831	1.09834	0.549172	−	0.835709i	$-0.314943\pi$
$577$	26.3831	1.09834	0.549172	+	0.835709i	$0.314943\pi$
$578$	0	0
$579$	−15.7587	−0.654909
$580$	0	0
$581$	29.7763	1.23533
$582$	0	0
$583$	−56.7735	−2.35132
$584$	0	0
$585$	−5.27337	−0.218027
$586$	0	0
$587$	2.22863	0.0919854	0.0459927	−	0.998942i	$-0.485355\pi$
$587$	2.22863	0.0919854	0.0459927	+	0.998942i	$0.485355\pi$
$588$	0	0
$589$	0.122235	0.00503661
$590$	0	0
$591$	50.8722	2.09260
$592$	0	0
$593$	38.5194	1.58180	0.790901	−	0.611945i	$-0.209612\pi$
$593$	38.5194	1.58180	0.790901	+	0.611945i	$0.209612\pi$
$594$	0	0
$595$	−16.7090	−0.685002
$596$	0	0
$597$	−20.2097	−0.827129
$598$	0	0
$599$	33.3227	1.36153	0.680764	−	0.732502i	$-0.261648\pi$
$599$	33.3227	1.36153	0.680764	+	0.732502i	$0.261648\pi$
$600$	0	0
$601$	−2.19653	−0.0895984	−0.0447992	−	0.998996i	$-0.514265\pi$
$601$	−2.19653	−0.0895984	−0.0447992	+	0.998996i	$0.514265\pi$
$602$	0	0
$603$	57.0539	2.32341
$604$	0	0
$605$	7.36264	0.299334
$606$	0	0
$607$	−22.9886	−0.933076	−0.466538	−	0.884501i	$-0.654499\pi$
$607$	−22.9886	−0.933076	−0.466538	+	0.884501i	$0.654499\pi$
$608$	0	0
$609$	88.8435	3.60012
$610$	0	0
$611$	4.33792	0.175494
$612$	0	0
$613$	6.53855	0.264090	0.132045	−	0.991244i	$-0.457846\pi$
$613$	6.53855	0.264090	0.132045	+	0.991244i	$0.457846\pi$
$614$	0	0
$615$	2.73123	0.110134
$616$	0	0
$617$	−32.4924	−1.30809	−0.654047	−	0.756454i	$-0.726930\pi$
$617$	−32.4924	−1.30809	−0.654047	+	0.756454i	$0.726930\pi$
$618$	0	0
$619$	24.6705	0.991592	0.495796	−	0.868439i	$-0.334876\pi$
$619$	24.6705	0.991592	0.495796	+	0.868439i	$0.334876\pi$
$620$	0	0
$621$	16.6791	0.669307
$622$	0	0
$623$	−4.52461	−0.181275
$624$	0	0
$625$	17.6122	0.704488
$626$	0	0
$627$	6.22096	0.248441
$628$	0	0
$629$	−70.8215	−2.82384
$630$	0	0
$631$	27.7676	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$631$	27.7676	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$632$	0	0
$633$	−60.8204	−2.41740
$634$	0	0
$635$	9.90470	0.393056
$636$	0	0
$637$	−14.6412	−0.580106
$638$	0	0
$639$	−4.83063	−0.191097
$640$	0	0
$641$	39.6937	1.56781	0.783903	−	0.620884i	$-0.213226\pi$
$641$	39.6937	1.56781	0.783903	+	0.620884i	$0.213226\pi$
$642$	0	0
$643$	−31.1296	−1.22763	−0.613816	−	0.789449i	$-0.710366\pi$
$643$	−31.1296	−1.22763	−0.613816	+	0.789449i	$0.710366\pi$
$644$	0	0
$645$	2.22875	0.0877568
$646$	0	0
$647$	−20.0933	−0.789950	−0.394975	−	0.918692i	$-0.629247\pi$
$647$	−20.0933	−0.789950	−0.394975	+	0.918692i	$0.629247\pi$
$648$	0	0
$649$	4.21954	0.165632
$650$	0	0
$651$	−2.55453	−0.100120
$652$	0	0
$653$	0.389868	0.0152567	0.00762835	−	0.999971i	$-0.497572\pi$
$653$	0.389868	0.0152567	0.00762835	+	0.999971i	$0.497572\pi$
$654$	0	0
$655$	14.6344	0.571812
$656$	0	0
$657$	−52.5688	−2.05091
$658$	0	0
$659$	−24.0522	−0.936942	−0.468471	−	0.883479i	$-0.655195\pi$
$659$	−24.0522	−0.936942	−0.468471	+	0.883479i	$0.655195\pi$
$660$	0	0
$661$	−38.1322	−1.48317	−0.741585	−	0.670859i	$-0.765926\pi$
$661$	−38.1322	−1.48317	−0.741585	+	0.670859i	$0.765926\pi$
$662$	0	0
$663$	−28.2315	−1.09642
$664$	0	0
$665$	1.40390	0.0544411
$666$	0	0
$667$	44.0080	1.70400
$668$	0	0
$669$	−74.3687	−2.87526
$670$	0	0
$671$	−17.8786	−0.690197
$672$	0	0
$673$	−4.81317	−0.185534	−0.0927670	−	0.995688i	$-0.529571\pi$
$673$	−4.81317	−0.185534	−0.0927670	+	0.995688i	$0.529571\pi$
$674$	0	0
$675$	14.4010	0.554295
$676$	0	0
$677$	−5.59186	−0.214913	−0.107456	−	0.994210i	$-0.534271\pi$
$677$	−5.59186	−0.214913	−0.107456	+	0.994210i	$0.534271\pi$
$678$	0	0
$679$	25.3759	0.973838
$680$	0	0
$681$	−7.45938	−0.285844
$682$	0	0
$683$	−23.4482	−0.897220	−0.448610	−	0.893728i	$-0.648081\pi$
$683$	−23.4482	−0.897220	−0.448610	+	0.893728i	$0.648081\pi$
$684$	0	0
$685$	4.86819	0.186004
$686$	0	0
$687$	−10.9587	−0.418100
$688$	0	0
$689$	−21.6476	−0.824708
$690$	0	0
$691$	−27.4172	−1.04300	−0.521499	−	0.853252i	$-0.674627\pi$
$691$	−27.4172	−1.04300	−0.521499	+	0.853252i	$0.674627\pi$
$692$	0	0
$693$	−75.8054	−2.87961
$694$	0	0
$695$	−0.320281	−0.0121489
$696$	0	0
$697$	8.52575	0.322936
$698$	0	0
$699$	−55.7685	−2.10936
$700$	0	0
$701$	2.06536	0.0780075	0.0390038	−	0.999239i	$-0.487582\pi$
$701$	2.06536	0.0780075	0.0390038	+	0.999239i	$0.487582\pi$
$702$	0	0
$703$	5.95048	0.224427
$704$	0	0
$705$	4.72029	0.177777
$706$	0	0
$707$	−14.0349	−0.527836
$708$	0	0
$709$	−28.6535	−1.07610	−0.538052	−	0.842911i	$-0.680840\pi$
$709$	−28.6535	−1.07610	−0.538052	+	0.842911i	$0.680840\pi$
$710$	0	0
$711$	17.8999	0.671298
$712$	0	0
$713$	−1.26537	−0.0473884
$714$	0	0
$715$	5.80223	0.216991
$716$	0	0
$717$	−45.8642	−1.71283
$718$	0	0
$719$	−9.94030	−0.370711	−0.185355	−	0.982672i	$-0.559344\pi$
$719$	−9.94030	−0.370711	−0.185355	+	0.982672i	$0.559344\pi$
$720$	0	0
$721$	54.2912	2.02191
$722$	0	0
$723$	61.6003	2.29094
$724$	0	0
$725$	37.9974	1.41119
$726$	0	0
$727$	−44.9252	−1.66618	−0.833092	−	0.553135i	$-0.813431\pi$
$727$	−44.9252	−1.66618	−0.833092	+	0.553135i	$0.813431\pi$
$728$	0	0
$729$	−42.5236	−1.57495
$730$	0	0
$731$	6.95721	0.257322
$732$	0	0
$733$	26.3989	0.975066	0.487533	−	0.873105i	$-0.337897\pi$
$733$	26.3989	0.975066	0.487533	+	0.873105i	$0.337897\pi$
$734$	0	0
$735$	−15.9318	−0.587653
$736$	0	0
$737$	−62.7758	−2.31238
$738$	0	0
$739$	12.4918	0.459519	0.229759	−	0.973247i	$-0.426206\pi$
$739$	12.4918	0.459519	0.229759	+	0.973247i	$0.426206\pi$
$740$	0	0
$741$	2.37204	0.0871389
$742$	0	0
$743$	45.4652	1.66796	0.833978	−	0.551797i	$-0.186058\pi$
$743$	45.4652	1.66796	0.833978	+	0.551797i	$0.186058\pi$
$744$	0	0
$745$	5.51178	0.201936
$746$	0	0
$747$	31.9205	1.16791
$748$	0	0
$749$	−25.0109	−0.913878
$750$	0	0
$751$	−31.9967	−1.16758	−0.583788	−	0.811906i	$-0.698430\pi$
$751$	−31.9967	−1.16758	−0.583788	+	0.811906i	$0.698430\pi$
$752$	0	0
$753$	43.2968	1.57782
$754$	0	0
$755$	−14.0826	−0.512517
$756$	0	0
$757$	6.13587	0.223012	0.111506	−	0.993764i	$-0.464433\pi$
$757$	6.13587	0.223012	0.111506	+	0.993764i	$0.464433\pi$
$758$	0	0
$759$	−64.3989	−2.33753
$760$	0	0
$761$	−27.0469	−0.980450	−0.490225	−	0.871596i	$-0.663085\pi$
$761$	−27.0469	−0.980450	−0.490225	+	0.871596i	$0.663085\pi$
$762$	0	0
$763$	16.7439	0.606169
$764$	0	0
$765$	−17.9122	−0.647618
$766$	0	0
$767$	1.60890	0.0580941
$768$	0	0
$769$	41.7673	1.50617	0.753083	−	0.657925i	$-0.228566\pi$
$769$	41.7673	1.50617	0.753083	+	0.657925i	$0.228566\pi$
$770$	0	0
$771$	15.4687	0.557091
$772$	0	0
$773$	32.3968	1.16523	0.582616	−	0.812747i	$-0.302029\pi$
$773$	32.3968	1.16523	0.582616	+	0.812747i	$0.302029\pi$
$774$	0	0
$775$	−1.09254	−0.0392453
$776$	0	0
$777$	−124.356	−4.46125
$778$	0	0
$779$	−0.716341	−0.0256656
$780$	0	0
$781$	5.31509	0.190189
$782$	0	0
$783$	27.1410	0.969939
$784$	0	0
$785$	−13.6091	−0.485729
$786$	0	0
$787$	−14.6914	−0.523691	−0.261845	−	0.965110i	$-0.584331\pi$
$787$	−14.6914	−0.523691	−0.261845	+	0.965110i	$0.584331\pi$
$788$	0	0
$789$	−12.3849	−0.440914
$790$	0	0
$791$	−63.2350	−2.24838
$792$	0	0
$793$	−6.81708	−0.242082
$794$	0	0
$795$	−23.5557	−0.835436
$796$	0	0
$797$	6.32441	0.224022	0.112011	−	0.993707i	$-0.464271\pi$
$797$	6.32441	0.224022	0.112011	+	0.993707i	$0.464271\pi$
$798$	0	0
$799$	14.7348	0.521279
$800$	0	0
$801$	−4.85044	−0.171382
$802$	0	0
$803$	57.8409	2.04116
$804$	0	0
$805$	−14.5331	−0.512225
$806$	0	0
$807$	−70.5716	−2.48424
$808$	0	0
$809$	32.4195	1.13981	0.569905	−	0.821711i	$-0.306980\pi$
$809$	32.4195	1.13981	0.569905	+	0.821711i	$0.306980\pi$
$810$	0	0
$811$	−13.9740	−0.490695	−0.245347	−	0.969435i	$-0.578902\pi$
$811$	−13.9740	−0.490695	−0.245347	+	0.969435i	$0.578902\pi$
$812$	0	0
$813$	7.05247	0.247341
$814$	0	0
$815$	3.90923	0.136934
$816$	0	0
$817$	−0.584551	−0.0204509
$818$	0	0
$819$	−28.9044	−1.01000
$820$	0	0
$821$	−50.8178	−1.77355	−0.886777	−	0.462198i	$-0.847061\pi$
$821$	−50.8178	−1.77355	−0.886777	+	0.462198i	$0.847061\pi$
$822$	0	0
$823$	−27.5649	−0.960851	−0.480425	−	0.877036i	$-0.659518\pi$
$823$	−27.5649	−0.960851	−0.480425	+	0.877036i	$0.659518\pi$
$824$	0	0
$825$	−55.6032	−1.93586
$826$	0	0
$827$	−29.7333	−1.03393	−0.516963	−	0.856008i	$-0.672938\pi$
$827$	−29.7333	−1.03393	−0.516963	+	0.856008i	$0.672938\pi$
$828$	0	0
$829$	20.3159	0.705601	0.352801	−	0.935699i	$-0.385229\pi$
$829$	20.3159	0.705601	0.352801	+	0.935699i	$0.385229\pi$
$830$	0	0
$831$	73.2382	2.54061
$832$	0	0
$833$	−49.7324	−1.72312
$834$	0	0
$835$	−9.69948	−0.335664
$836$	0	0
$837$	−0.780387	−0.0269741
$838$	0	0
$839$	−38.4414	−1.32714	−0.663572	−	0.748113i	$-0.730960\pi$
$839$	−38.4414	−1.32714	−0.663572	+	0.748113i	$0.730960\pi$
$840$	0	0
$841$	42.6120	1.46938
$842$	0	0
$843$	73.0624	2.51640
$844$	0	0
$845$	−7.07012	−0.243220
$846$	0	0
$847$	40.3562	1.38666
$848$	0	0
$849$	61.8278	2.12193
$850$	0	0
$851$	−61.5989	−2.11158
$852$	0	0
$853$	29.8021	1.02040	0.510202	−	0.860055i	$-0.329571\pi$
$853$	29.8021	1.02040	0.510202	+	0.860055i	$0.329571\pi$
$854$	0	0
$855$	1.50500	0.0514700
$856$	0	0
$857$	28.3198	0.967387	0.483694	−	0.875237i	$-0.339295\pi$
$857$	28.3198	0.967387	0.483694	+	0.875237i	$0.339295\pi$
$858$	0	0
$859$	−7.26805	−0.247983	−0.123991	−	0.992283i	$-0.539569\pi$
$859$	−7.26805	−0.247983	−0.123991	+	0.992283i	$0.539569\pi$
$860$	0	0
$861$	14.9704	0.510191
$862$	0	0
$863$	17.5757	0.598282	0.299141	−	0.954209i	$-0.403300\pi$
$863$	17.5757	0.598282	0.299141	+	0.954209i	$0.403300\pi$
$864$	0	0
$865$	17.0893	0.581054
$866$	0	0
$867$	−50.2930	−1.70804
$868$	0	0
$869$	−19.6951	−0.668109
$870$	0	0
$871$	−23.9363	−0.811050
$872$	0	0
$873$	27.2033	0.920691
$874$	0	0
$875$	−26.5212	−0.896579
$876$	0	0
$877$	−6.75635	−0.228146	−0.114073	−	0.993472i	$-0.536390\pi$
$877$	−6.75635	−0.228146	−0.114073	+	0.993472i	$0.536390\pi$
$878$	0	0
$879$	29.6591	1.00038
$880$	0	0
$881$	2.77043	0.0933383	0.0466692	−	0.998910i	$-0.485139\pi$
$881$	2.77043	0.0933383	0.0466692	+	0.998910i	$0.485139\pi$
$882$	0	0
$883$	−0.275545	−0.00927281	−0.00463641	−	0.999989i	$-0.501476\pi$
$883$	−0.275545	−0.00927281	−0.00463641	+	0.999989i	$0.501476\pi$
$884$	0	0
$885$	1.75072	0.0588498
$886$	0	0
$887$	−8.98744	−0.301769	−0.150884	−	0.988551i	$-0.548212\pi$
$887$	−8.98744	−0.301769	−0.150884	+	0.988551i	$0.548212\pi$
$888$	0	0
$889$	54.2898	1.82082
$890$	0	0
$891$	18.3898	0.616080
$892$	0	0
$893$	−1.23803	−0.0414291
$894$	0	0
$895$	−0.386331	−0.0129136
$896$	0	0
$897$	−24.5551	−0.819872
$898$	0	0
$899$	−2.05907	−0.0686738
$900$	0	0
$901$	−73.5312	−2.44968
$902$	0	0
$903$	12.2162	0.406531
$904$	0	0
$905$	17.7121	0.588772
$906$	0	0
$907$	30.7323	1.02045	0.510225	−	0.860041i	$-0.329562\pi$
$907$	30.7323	1.02045	0.510225	+	0.860041i	$0.329562\pi$
$908$	0	0
$909$	−15.0455	−0.499029
$910$	0	0
$911$	−14.5886	−0.483340	−0.241670	−	0.970358i	$-0.577695\pi$
$911$	−14.5886	−0.483340	−0.241670	+	0.970358i	$0.577695\pi$
$912$	0	0
$913$	−35.1218	−1.16236
$914$	0	0
$915$	−7.41798	−0.245231
$916$	0	0
$917$	80.2141	2.64890
$918$	0	0
$919$	9.16059	0.302180	0.151090	−	0.988520i	$-0.451722\pi$
$919$	9.16059	0.302180	0.151090	+	0.988520i	$0.451722\pi$
$920$	0	0
$921$	−25.5799	−0.842886
$922$	0	0
$923$	2.02663	0.0667075
$924$	0	0
$925$	−53.1857	−1.74874
$926$	0	0
$927$	58.2008	1.91156
$928$	0	0
$929$	−20.6834	−0.678600	−0.339300	−	0.940678i	$-0.610190\pi$
$929$	−20.6834	−0.678600	−0.339300	+	0.940678i	$0.610190\pi$
$930$	0	0
$931$	4.17856	0.136947
$932$	0	0
$933$	−88.8069	−2.90741
$934$	0	0
$935$	19.7086	0.644542
$936$	0	0
$937$	54.1811	1.77002	0.885009	−	0.465574i	$-0.154152\pi$
$937$	54.1811	1.77002	0.885009	+	0.465574i	$0.154152\pi$
$938$	0	0
$939$	−54.5429	−1.77994
$940$	0	0
$941$	29.8816	0.974113	0.487056	−	0.873370i	$-0.338071\pi$
$941$	29.8816	0.974113	0.487056	+	0.873370i	$0.338071\pi$
$942$	0	0
$943$	7.41551	0.241482
$944$	0	0
$945$	−8.96297	−0.291566
$946$	0	0
$947$	−11.9099	−0.387021	−0.193510	−	0.981098i	$-0.561987\pi$
$947$	−11.9099	−0.387021	−0.193510	+	0.981098i	$0.561987\pi$
$948$	0	0
$949$	22.0546	0.715923
$950$	0	0
$951$	69.4126	2.25086
$952$	0	0
$953$	46.5110	1.50664	0.753320	−	0.657655i	$-0.228451\pi$
$953$	46.5110	1.50664	0.753320	+	0.657655i	$0.228451\pi$
$954$	0	0
$955$	14.6264	0.473301
$956$	0	0
$957$	−104.793	−3.38747
$958$	0	0
$959$	26.6836	0.861659
$960$	0	0
$961$	−30.9408	−0.998090
$962$	0	0
$963$	−26.8120	−0.864004
$964$	0	0
$965$	4.19476	0.135034
$966$	0	0
$967$	−22.6297	−0.727723	−0.363861	−	0.931453i	$-0.618542\pi$
$967$	−22.6297	−0.727723	−0.363861	+	0.931453i	$0.618542\pi$
$968$	0	0
$969$	8.05718	0.258834
$970$	0	0
$971$	−1.88229	−0.0604056	−0.0302028	−	0.999544i	$-0.509615\pi$
$971$	−1.88229	−0.0604056	−0.0302028	+	0.999544i	$0.509615\pi$
$972$	0	0
$973$	−1.75553	−0.0562796
$974$	0	0
$975$	−21.2014	−0.678988
$976$	0	0
$977$	8.35601	0.267332	0.133666	−	0.991026i	$-0.457325\pi$
$977$	8.35601	0.267332	0.133666	+	0.991026i	$0.457325\pi$
$978$	0	0
$979$	5.33689	0.170568
$980$	0	0
$981$	17.9496	0.573088
$982$	0	0
$983$	−13.0342	−0.415727	−0.207863	−	0.978158i	$-0.566651\pi$
$983$	−13.0342	−0.415727	−0.207863	+	0.978158i	$0.566651\pi$
$984$	0	0
$985$	−13.5415	−0.431469
$986$	0	0
$987$	25.8729	0.823545
$988$	0	0
$989$	6.05123	0.192418
$990$	0	0
$991$	−56.6071	−1.79818	−0.899091	−	0.437761i	$-0.855772\pi$
$991$	−56.6071	−1.79818	−0.899091	+	0.437761i	$0.855772\pi$
$992$	0	0
$993$	−64.9724	−2.06184
$994$	0	0
$995$	5.37957	0.170544
$996$	0	0
$997$	5.45902	0.172889	0.0864445	−	0.996257i	$-0.472449\pi$
$997$	5.45902	0.172889	0.0864445	+	0.996257i	$0.472449\pi$
$998$	0	0
$999$	−37.9898	−1.20194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.6	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-2.68247 q^{3} +0.714038 q^{5} +3.91380 q^{7} +4.19563 q^{9} +O(q^{10})\)	\(q-2.68247 q^{3} +0.714038 q^{5} +3.91380 q^{7} +4.19563 q^{9} -4.61641 q^{11} -1.76023 q^{13} -1.91538 q^{15} -5.97903 q^{17} +0.502363 q^{19} -10.4986 q^{21} -5.20043 q^{23} -4.49015 q^{25} -3.20725 q^{27} -8.46239 q^{29} +0.243320 q^{31} +12.3834 q^{33} +2.79460 q^{35} +11.8450 q^{37} +4.72175 q^{39} -1.42594 q^{41} -1.16360 q^{43} +2.99584 q^{45} -2.46441 q^{47} +8.31780 q^{49} +16.0386 q^{51} +12.2982 q^{53} -3.29630 q^{55} -1.34757 q^{57} -0.914030 q^{59} +3.87284 q^{61} +16.4209 q^{63} -1.25687 q^{65} +13.5984 q^{67} +13.9500 q^{69} -1.15135 q^{71} -12.5294 q^{73} +12.0447 q^{75} -18.0677 q^{77} +4.26631 q^{79} -3.98356 q^{81} +7.60803 q^{83} -4.26926 q^{85} +22.7001 q^{87} -1.15607 q^{89} -6.88917 q^{91} -0.652698 q^{93} +0.358707 q^{95} +6.48371 q^{97} -19.3688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

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