LMFDB - Embedded newform 6044.2.a.b.1.15

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			1.15
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)

$f(q)$	$=$	$q-2.06433 q^{3} +4.18134 q^{5} +5.19775 q^{7} +1.26148 q^{9} +O(q^{10})$	$q-2.06433 q^{3} +4.18134 q^{5} +5.19775 q^{7} +1.26148 q^{9} +1.40281 q^{11} +1.87181 q^{13} -8.63168 q^{15} +0.798178 q^{17} +8.00011 q^{19} -10.7299 q^{21} -3.11598 q^{23} +12.4836 q^{25} +3.58889 q^{27} -1.44716 q^{29} -8.36943 q^{31} -2.89587 q^{33} +21.7335 q^{35} -5.38214 q^{37} -3.86403 q^{39} -0.673670 q^{41} +3.47805 q^{43} +5.27466 q^{45} +9.22654 q^{47} +20.0166 q^{49} -1.64771 q^{51} -1.77972 q^{53} +5.86563 q^{55} -16.5149 q^{57} -9.85004 q^{59} -3.04932 q^{61} +6.55683 q^{63} +7.82666 q^{65} +8.09074 q^{67} +6.43242 q^{69} -4.93386 q^{71} +12.1005 q^{73} -25.7703 q^{75} +7.29145 q^{77} +10.4265 q^{79} -11.1931 q^{81} +9.78473 q^{83} +3.33745 q^{85} +2.98743 q^{87} +3.10939 q^{89} +9.72917 q^{91} +17.2773 q^{93} +33.4512 q^{95} -9.85245 q^{97} +1.76961 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.06433	−1.19184	−0.595922	−	0.803042i	$-0.703213\pi$
$3$	−2.06433	−1.19184	−0.595922	+	0.803042i	$0.703213\pi$
$4$	0	0
$5$	4.18134	1.86995	0.934976	−	0.354711i	$-0.115421\pi$
$5$	4.18134	1.86995	0.934976	+	0.354711i	$0.115421\pi$
$6$	0	0
$7$	5.19775	1.96456	0.982282	−	0.187410i	$-0.0600094\pi$
$7$	5.19775	1.96456	0.982282	+	0.187410i	$0.0600094\pi$
$8$	0	0
$9$	1.26148	0.420492
$10$	0	0
$11$	1.40281	0.422963	0.211482	−	0.977382i	$-0.432171\pi$
$11$	1.40281	0.422963	0.211482	+	0.977382i	$0.432171\pi$
$12$	0	0
$13$	1.87181	0.519145	0.259573	−	0.965724i	$-0.416418\pi$
$13$	1.87181	0.519145	0.259573	+	0.965724i	$0.416418\pi$
$14$	0	0
$15$	−8.63168	−2.22869
$16$	0	0
$17$	0.798178	0.193587	0.0967933	−	0.995305i	$-0.469141\pi$
$17$	0.798178	0.193587	0.0967933	+	0.995305i	$0.469141\pi$
$18$	0	0
$19$	8.00011	1.83535	0.917675	−	0.397331i	$-0.130064\pi$
$19$	8.00011	1.83535	0.917675	+	0.397331i	$0.130064\pi$
$20$	0	0
$21$	−10.7299	−2.34145
$22$	0	0
$23$	−3.11598	−0.649727	−0.324863	−	0.945761i	$-0.605318\pi$
$23$	−3.11598	−0.649727	−0.324863	+	0.945761i	$0.605318\pi$
$24$	0	0
$25$	12.4836	2.49672
$26$	0	0
$27$	3.58889	0.690683
$28$	0	0
$29$	−1.44716	−0.268732	−0.134366	−	0.990932i	$-0.542900\pi$
$29$	−1.44716	−0.268732	−0.134366	+	0.990932i	$0.542900\pi$
$30$	0	0
$31$	−8.36943	−1.50319	−0.751597	−	0.659623i	$-0.770716\pi$
$31$	−8.36943	−1.50319	−0.751597	+	0.659623i	$0.770716\pi$
$32$	0	0
$33$	−2.89587	−0.504106
$34$	0	0
$35$	21.7335	3.67364
$36$	0	0
$37$	−5.38214	−0.884818	−0.442409	−	0.896813i	$-0.645876\pi$
$37$	−5.38214	−0.884818	−0.442409	+	0.896813i	$0.645876\pi$
$38$	0	0
$39$	−3.86403	−0.618740
$40$	0	0
$41$	−0.673670	−0.105210	−0.0526048	−	0.998615i	$-0.516752\pi$
$41$	−0.673670	−0.105210	−0.0526048	+	0.998615i	$0.516752\pi$
$42$	0	0
$43$	3.47805	0.530397	0.265199	−	0.964194i	$-0.414562\pi$
$43$	3.47805	0.530397	0.265199	+	0.964194i	$0.414562\pi$
$44$	0	0
$45$	5.27466	0.786300
$46$	0	0
$47$	9.22654	1.34583	0.672914	−	0.739720i	$-0.265042\pi$
$47$	9.22654	1.34583	0.672914	+	0.739720i	$0.265042\pi$
$48$	0	0
$49$	20.0166	2.85951
$50$	0	0
$51$	−1.64771	−0.230725
$52$	0	0
$53$	−1.77972	−0.244463	−0.122232	−	0.992502i	$-0.539005\pi$
$53$	−1.77972	−0.244463	−0.122232	+	0.992502i	$0.539005\pi$
$54$	0	0
$55$	5.86563	0.790921
$56$	0	0
$57$	−16.5149	−2.18745
$58$	0	0
$59$	−9.85004	−1.28237	−0.641183	−	0.767388i	$-0.721556\pi$
$59$	−9.85004	−1.28237	−0.641183	+	0.767388i	$0.721556\pi$
$60$	0	0
$61$	−3.04932	−0.390426	−0.195213	−	0.980761i	$-0.562540\pi$
$61$	−3.04932	−0.390426	−0.195213	+	0.980761i	$0.562540\pi$
$62$	0	0
$63$	6.55683	0.826083
$64$	0	0
$65$	7.82666	0.970777
$66$	0	0
$67$	8.09074	0.988441	0.494221	−	0.869336i	$-0.335453\pi$
$67$	8.09074	0.988441	0.494221	+	0.869336i	$0.335453\pi$
$68$	0	0
$69$	6.43242	0.774373
$70$	0	0
$71$	−4.93386	−0.585541	−0.292771	−	0.956183i	$-0.594577\pi$
$71$	−4.93386	−0.585541	−0.292771	+	0.956183i	$0.594577\pi$
$72$	0	0
$73$	12.1005	1.41625	0.708125	−	0.706087i	$-0.249541\pi$
$73$	12.1005	1.41625	0.708125	+	0.706087i	$0.249541\pi$
$74$	0	0
$75$	−25.7703	−2.97570
$76$	0	0
$77$	7.29145	0.830938
$78$	0	0
$79$	10.4265	1.17308	0.586538	−	0.809922i	$-0.300491\pi$
$79$	10.4265	1.17308	0.586538	+	0.809922i	$0.300491\pi$
$80$	0	0
$81$	−11.1931	−1.24368
$82$	0	0
$83$	9.78473	1.07401	0.537007	−	0.843578i	$-0.319555\pi$
$83$	9.78473	1.07401	0.537007	+	0.843578i	$0.319555\pi$
$84$	0	0
$85$	3.33745	0.361998
$86$	0	0
$87$	2.98743	0.320286
$88$	0	0
$89$	3.10939	0.329595	0.164797	−	0.986327i	$-0.447303\pi$
$89$	3.10939	0.329595	0.164797	+	0.986327i	$0.447303\pi$
$90$	0	0
$91$	9.72917	1.01989
$92$	0	0
$93$	17.2773	1.79157
$94$	0	0
$95$	33.4512	3.43202
$96$	0	0
$97$	−9.85245	−1.00036	−0.500182	−	0.865920i	$-0.666734\pi$
$97$	−9.85245	−1.00036	−0.500182	+	0.865920i	$0.666734\pi$
$98$	0	0
$99$	1.76961	0.177853
$100$	0	0
$101$	−8.52739	−0.848507	−0.424253	−	0.905543i	$-0.639463\pi$
$101$	−8.52739	−0.848507	−0.424253	+	0.905543i	$0.639463\pi$
$102$	0	0
$103$	−2.79325	−0.275227	−0.137614	−	0.990486i	$-0.543943\pi$
$103$	−2.79325	−0.275227	−0.137614	+	0.990486i	$0.543943\pi$
$104$	0	0
$105$	−44.8653	−4.37841
$106$	0	0
$107$	−9.31063	−0.900092	−0.450046	−	0.893005i	$-0.648592\pi$
$107$	−9.31063	−0.900092	−0.450046	+	0.893005i	$0.648592\pi$
$108$	0	0
$109$	−7.61520	−0.729404	−0.364702	−	0.931124i	$-0.618829\pi$
$109$	−7.61520	−0.729404	−0.364702	+	0.931124i	$0.618829\pi$
$110$	0	0
$111$	11.1105	1.05457
$112$	0	0
$113$	−3.14541	−0.295896	−0.147948	−	0.988995i	$-0.547267\pi$
$113$	−3.14541	−0.295896	−0.147948	+	0.988995i	$0.547267\pi$
$114$	0	0
$115$	−13.0290	−1.21496
$116$	0	0
$117$	2.36124	0.218296
$118$	0	0
$119$	4.14873	0.380313
$120$	0	0
$121$	−9.03212	−0.821102
$122$	0	0
$123$	1.39068	0.125393
$124$	0	0
$125$	31.2915	2.79880
$126$	0	0
$127$	−8.12382	−0.720872	−0.360436	−	0.932784i	$-0.617372\pi$
$127$	−8.12382	−0.720872	−0.360436	+	0.932784i	$0.617372\pi$
$128$	0	0
$129$	−7.17985	−0.632151
$130$	0	0
$131$	−12.1866	−1.06475	−0.532374	−	0.846509i	$-0.678700\pi$
$131$	−12.1866	−1.06475	−0.532374	+	0.846509i	$0.678700\pi$
$132$	0	0
$133$	41.5825	3.60566
$134$	0	0
$135$	15.0064	1.29154
$136$	0	0
$137$	−12.1611	−1.03899	−0.519495	−	0.854473i	$-0.673880\pi$
$137$	−12.1611	−1.03899	−0.519495	+	0.854473i	$0.673880\pi$
$138$	0	0
$139$	−1.13432	−0.0962117	−0.0481059	−	0.998842i	$-0.515318\pi$
$139$	−1.13432	−0.0962117	−0.0481059	+	0.998842i	$0.515318\pi$
$140$	0	0
$141$	−19.0467	−1.60402
$142$	0	0
$143$	2.62579	0.219579
$144$	0	0
$145$	−6.05108	−0.502515
$146$	0	0
$147$	−41.3209	−3.40809
$148$	0	0
$149$	−12.4131	−1.01692	−0.508461	−	0.861085i	$-0.669785\pi$
$149$	−12.4131	−1.01692	−0.508461	+	0.861085i	$0.669785\pi$
$150$	0	0
$151$	14.8812	1.21102	0.605509	−	0.795839i	$-0.292970\pi$
$151$	14.8812	1.21102	0.605509	+	0.795839i	$0.292970\pi$
$152$	0	0
$153$	1.00688	0.0814016
$154$	0	0
$155$	−34.9954	−2.81090
$156$	0	0
$157$	−15.7066	−1.25352	−0.626760	−	0.779212i	$-0.715619\pi$
$157$	−15.7066	−1.25352	−0.626760	+	0.779212i	$0.715619\pi$
$158$	0	0
$159$	3.67393	0.291362
$160$	0	0
$161$	−16.1961	−1.27643
$162$	0	0
$163$	−1.82040	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$163$	−1.82040	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$164$	0	0
$165$	−12.1086	−0.942654
$166$	0	0
$167$	−3.61714	−0.279903	−0.139951	−	0.990158i	$-0.544695\pi$
$167$	−3.61714	−0.279903	−0.139951	+	0.990158i	$0.544695\pi$
$168$	0	0
$169$	−9.49634	−0.730488
$170$	0	0
$171$	10.0919	0.771750
$172$	0	0
$173$	−8.19684	−0.623194	−0.311597	−	0.950214i	$-0.600864\pi$
$173$	−8.19684	−0.623194	−0.311597	+	0.950214i	$0.600864\pi$
$174$	0	0
$175$	64.8866	4.90497
$176$	0	0
$177$	20.3338	1.52838
$178$	0	0
$179$	−19.5555	−1.46164	−0.730822	−	0.682568i	$-0.760863\pi$
$179$	−19.5555	−1.46164	−0.730822	+	0.682568i	$0.760863\pi$
$180$	0	0
$181$	2.18677	0.162542	0.0812708	−	0.996692i	$-0.474102\pi$
$181$	2.18677	0.162542	0.0812708	+	0.996692i	$0.474102\pi$
$182$	0	0
$183$	6.29482	0.465327
$184$	0	0
$185$	−22.5046	−1.65457
$186$	0	0
$187$	1.11969	0.0818800
$188$	0	0
$189$	18.6542	1.35689
$190$	0	0
$191$	20.6601	1.49491	0.747457	−	0.664310i	$-0.231274\pi$
$191$	20.6601	1.49491	0.747457	+	0.664310i	$0.231274\pi$
$192$	0	0
$193$	−11.4483	−0.824064	−0.412032	−	0.911169i	$-0.635181\pi$
$193$	−11.4483	−0.824064	−0.412032	+	0.911169i	$0.635181\pi$
$194$	0	0
$195$	−16.1568	−1.15701
$196$	0	0
$197$	−23.0165	−1.63986	−0.819930	−	0.572463i	$-0.805988\pi$
$197$	−23.0165	−1.63986	−0.819930	+	0.572463i	$0.805988\pi$
$198$	0	0
$199$	3.50757	0.248645	0.124323	−	0.992242i	$-0.460324\pi$
$199$	3.50757	0.248645	0.124323	+	0.992242i	$0.460324\pi$
$200$	0	0
$201$	−16.7020	−1.17807
$202$	0	0
$203$	−7.52199	−0.527940
$204$	0	0
$205$	−2.81684	−0.196737
$206$	0	0
$207$	−3.93073	−0.273205
$208$	0	0
$209$	11.2226	0.776286
$210$	0	0
$211$	14.2433	0.980548	0.490274	−	0.871568i	$-0.336897\pi$
$211$	14.2433	0.980548	0.490274	+	0.871568i	$0.336897\pi$
$212$	0	0
$213$	10.1851	0.697874
$214$	0	0
$215$	14.5429	0.991817
$216$	0	0
$217$	−43.5022	−2.95312
$218$	0	0
$219$	−24.9794	−1.68795
$220$	0	0
$221$	1.49403	0.100500
$222$	0	0
$223$	2.83006	0.189515	0.0947573	−	0.995500i	$-0.469792\pi$
$223$	2.83006	0.189515	0.0947573	+	0.995500i	$0.469792\pi$
$224$	0	0
$225$	15.7478	1.04985
$226$	0	0
$227$	14.1592	0.939777	0.469888	−	0.882726i	$-0.344294\pi$
$227$	14.1592	0.939777	0.469888	+	0.882726i	$0.344294\pi$
$228$	0	0
$229$	0.495726	0.0327585	0.0163793	−	0.999866i	$-0.494786\pi$
$229$	0.495726	0.0327585	0.0163793	+	0.999866i	$0.494786\pi$
$230$	0	0
$231$	−15.0520	−0.990348
$232$	0	0
$233$	6.82431	0.447076	0.223538	−	0.974695i	$-0.428239\pi$
$233$	6.82431	0.447076	0.223538	+	0.974695i	$0.428239\pi$
$234$	0	0
$235$	38.5793	2.51664
$236$	0	0
$237$	−21.5238	−1.39812
$238$	0	0
$239$	−21.1463	−1.36784	−0.683919	−	0.729558i	$-0.739726\pi$
$239$	−21.1463	−1.36784	−0.683919	+	0.729558i	$0.739726\pi$
$240$	0	0
$241$	−6.82486	−0.439628	−0.219814	−	0.975542i	$-0.570545\pi$
$241$	−6.82486	−0.439628	−0.219814	+	0.975542i	$0.570545\pi$
$242$	0	0
$243$	12.3396	0.791588
$244$	0	0
$245$	83.6961	5.34715
$246$	0	0
$247$	14.9746	0.952814
$248$	0	0
$249$	−20.1990	−1.28006
$250$	0	0
$251$	8.42174	0.531576	0.265788	−	0.964032i	$-0.414368\pi$
$251$	8.42174	0.531576	0.265788	+	0.964032i	$0.414368\pi$
$252$	0	0
$253$	−4.37113	−0.274810
$254$	0	0
$255$	−6.88962	−0.431445
$256$	0	0
$257$	25.0487	1.56250	0.781248	−	0.624220i	$-0.214583\pi$
$257$	25.0487	1.56250	0.781248	+	0.624220i	$0.214583\pi$
$258$	0	0
$259$	−27.9750	−1.73828
$260$	0	0
$261$	−1.82556	−0.112999
$262$	0	0
$263$	−23.1375	−1.42672	−0.713360	−	0.700798i	$-0.752827\pi$
$263$	−23.1375	−1.42672	−0.713360	+	0.700798i	$0.752827\pi$
$264$	0	0
$265$	−7.44161	−0.457134
$266$	0	0
$267$	−6.41882	−0.392826
$268$	0	0
$269$	−22.4373	−1.36802	−0.684012	−	0.729471i	$-0.739766\pi$
$269$	−22.4373	−1.36802	−0.684012	+	0.729471i	$0.739766\pi$
$270$	0	0
$271$	26.2402	1.59398	0.796990	−	0.603992i	$-0.206424\pi$
$271$	26.2402	1.59398	0.796990	+	0.603992i	$0.206424\pi$
$272$	0	0
$273$	−20.0843	−1.21555
$274$	0	0
$275$	17.5121	1.05602
$276$	0	0
$277$	−22.5528	−1.35507	−0.677533	−	0.735493i	$-0.736951\pi$
$277$	−22.5528	−1.35507	−0.677533	+	0.735493i	$0.736951\pi$
$278$	0	0
$279$	−10.5578	−0.632081
$280$	0	0
$281$	−7.28557	−0.434621	−0.217310	−	0.976103i	$-0.569728\pi$
$281$	−7.28557	−0.434621	−0.217310	+	0.976103i	$0.569728\pi$
$282$	0	0
$283$	12.8482	0.763744	0.381872	−	0.924215i	$-0.375279\pi$
$283$	12.8482	0.763744	0.381872	+	0.924215i	$0.375279\pi$
$284$	0	0
$285$	−69.0544	−4.09043
$286$	0	0
$287$	−3.50156	−0.206691
$288$	0	0
$289$	−16.3629	−0.962524
$290$	0	0
$291$	20.3388	1.19228
$292$	0	0
$293$	26.7147	1.56069	0.780344	−	0.625350i	$-0.215044\pi$
$293$	26.7147	1.56069	0.780344	+	0.625350i	$0.215044\pi$
$294$	0	0
$295$	−41.1864	−2.39796
$296$	0	0
$297$	5.03454	0.292134
$298$	0	0
$299$	−5.83251	−0.337303
$300$	0	0
$301$	18.0780	1.04200
$302$	0	0
$303$	17.6034	1.01129
$304$	0	0
$305$	−12.7503	−0.730078
$306$	0	0
$307$	−0.210874	−0.0120352	−0.00601761	−	0.999982i	$-0.501915\pi$
$307$	−0.210874	−0.0120352	−0.00601761	+	0.999982i	$0.501915\pi$
$308$	0	0
$309$	5.76621	0.328028
$310$	0	0
$311$	−14.8898	−0.844323	−0.422161	−	0.906521i	$-0.638728\pi$
$311$	−14.8898	−0.844323	−0.422161	+	0.906521i	$0.638728\pi$
$312$	0	0
$313$	−27.7258	−1.56715	−0.783577	−	0.621295i	$-0.786607\pi$
$313$	−27.7258	−1.56715	−0.783577	+	0.621295i	$0.786607\pi$
$314$	0	0
$315$	27.4163	1.54474
$316$	0	0
$317$	−16.7785	−0.942373	−0.471187	−	0.882034i	$-0.656174\pi$
$317$	−16.7785	−0.942373	−0.471187	+	0.882034i	$0.656174\pi$
$318$	0	0
$319$	−2.03010	−0.113664
$320$	0	0
$321$	19.2202	1.07277
$322$	0	0
$323$	6.38551	0.355299
$324$	0	0
$325$	23.3669	1.29616
$326$	0	0
$327$	15.7203	0.869335
$328$	0	0
$329$	47.9572	2.64397
$330$	0	0
$331$	8.78065	0.482628	0.241314	−	0.970447i	$-0.422422\pi$
$331$	8.78065	0.482628	0.241314	+	0.970447i	$0.422422\pi$
$332$	0	0
$333$	−6.78944	−0.372059
$334$	0	0
$335$	33.8301	1.84834
$336$	0	0
$337$	24.0111	1.30797	0.653983	−	0.756509i	$-0.273097\pi$
$337$	24.0111	1.30797	0.653983	+	0.756509i	$0.273097\pi$
$338$	0	0
$339$	6.49319	0.352661
$340$	0	0
$341$	−11.7407	−0.635795
$342$	0	0
$343$	67.6568	3.65312
$344$	0	0
$345$	26.8961	1.44804
$346$	0	0
$347$	29.9486	1.60772	0.803862	−	0.594816i	$-0.202775\pi$
$347$	29.9486	1.60772	0.803862	+	0.594816i	$0.202775\pi$
$348$	0	0
$349$	8.82583	0.472436	0.236218	−	0.971700i	$-0.424092\pi$
$349$	8.82583	0.472436	0.236218	+	0.971700i	$0.424092\pi$
$350$	0	0
$351$	6.71771	0.358565
$352$	0	0
$353$	18.8121	1.00127	0.500634	−	0.865659i	$-0.333100\pi$
$353$	18.8121	1.00127	0.500634	+	0.865659i	$0.333100\pi$
$354$	0	0
$355$	−20.6301	−1.09493
$356$	0	0
$357$	−8.56436	−0.453274
$358$	0	0
$359$	11.1093	0.586329	0.293164	−	0.956062i	$-0.405292\pi$
$359$	11.1093	0.586329	0.293164	+	0.956062i	$0.405292\pi$
$360$	0	0
$361$	45.0017	2.36851
$362$	0	0
$363$	18.6453	0.978626
$364$	0	0
$365$	50.5961	2.64832
$366$	0	0
$367$	−27.0248	−1.41068	−0.705340	−	0.708869i	$-0.749206\pi$
$367$	−27.0248	−1.41068	−0.705340	+	0.708869i	$0.749206\pi$
$368$	0	0
$369$	−0.849818	−0.0442398
$370$	0	0
$371$	−9.25053	−0.480263
$372$	0	0
$373$	20.0480	1.03805	0.519024	−	0.854760i	$-0.326296\pi$
$373$	20.0480	1.03805	0.519024	+	0.854760i	$0.326296\pi$
$374$	0	0
$375$	−64.5961	−3.33573
$376$	0	0
$377$	−2.70881	−0.139511
$378$	0	0
$379$	−16.4047	−0.842654	−0.421327	−	0.906909i	$-0.638435\pi$
$379$	−16.4047	−0.842654	−0.421327	+	0.906909i	$0.638435\pi$
$380$	0	0
$381$	16.7703	0.859167
$382$	0	0
$383$	−9.62326	−0.491725	−0.245863	−	0.969305i	$-0.579071\pi$
$383$	−9.62326	−0.491725	−0.245863	+	0.969305i	$0.579071\pi$
$384$	0	0
$385$	30.4880	1.55381
$386$	0	0
$387$	4.38747	0.223028
$388$	0	0
$389$	9.91044	0.502479	0.251240	−	0.967925i	$-0.419162\pi$
$389$	9.91044	0.502479	0.251240	+	0.967925i	$0.419162\pi$
$390$	0	0
$391$	−2.48711	−0.125778
$392$	0	0
$393$	25.1572	1.26901
$394$	0	0
$395$	43.5968	2.19360
$396$	0	0
$397$	34.9633	1.75476	0.877379	−	0.479797i	$-0.159290\pi$
$397$	34.9633	1.75476	0.877379	+	0.479797i	$0.159290\pi$
$398$	0	0
$399$	−85.8402	−4.29739
$400$	0	0
$401$	−23.9659	−1.19680	−0.598399	−	0.801198i	$-0.704196\pi$
$401$	−23.9659	−1.19680	−0.598399	+	0.801198i	$0.704196\pi$
$402$	0	0
$403$	−15.6659	−0.780376
$404$	0	0
$405$	−46.8022	−2.32562
$406$	0	0
$407$	−7.55012	−0.374245
$408$	0	0
$409$	15.0403	0.743696	0.371848	−	0.928294i	$-0.378724\pi$
$409$	15.0403	0.743696	0.371848	+	0.928294i	$0.378724\pi$
$410$	0	0
$411$	25.1045	1.23831
$412$	0	0
$413$	−51.1980	−2.51929
$414$	0	0
$415$	40.9133	2.00835
$416$	0	0
$417$	2.34161	0.114669
$418$	0	0
$419$	12.0992	0.591085	0.295543	−	0.955330i	$-0.404500\pi$
$419$	12.0992	0.591085	0.295543	+	0.955330i	$0.404500\pi$
$420$	0	0
$421$	38.5137	1.87705	0.938523	−	0.345218i	$-0.112195\pi$
$421$	38.5137	1.87705	0.938523	+	0.345218i	$0.112195\pi$
$422$	0	0
$423$	11.6391	0.565910
$424$	0	0
$425$	9.96414	0.483332
$426$	0	0
$427$	−15.8496	−0.767016
$428$	0	0
$429$	−5.42050	−0.261704
$430$	0	0
$431$	−30.3060	−1.45979	−0.729893	−	0.683561i	$-0.760430\pi$
$431$	−30.3060	−1.45979	−0.729893	+	0.683561i	$0.760430\pi$
$432$	0	0
$433$	−29.3410	−1.41004	−0.705020	−	0.709187i	$-0.749062\pi$
$433$	−29.3410	−1.41004	−0.705020	+	0.709187i	$0.749062\pi$
$434$	0	0
$435$	12.4915	0.598920
$436$	0	0
$437$	−24.9282	−1.19248
$438$	0	0
$439$	28.3021	1.35079	0.675393	−	0.737458i	$-0.263974\pi$
$439$	28.3021	1.35079	0.675393	+	0.737458i	$0.263974\pi$
$440$	0	0
$441$	25.2504	1.20240
$442$	0	0
$443$	6.59896	0.313526	0.156763	−	0.987636i	$-0.449894\pi$
$443$	6.59896	0.313526	0.156763	+	0.987636i	$0.449894\pi$
$444$	0	0
$445$	13.0014	0.616327
$446$	0	0
$447$	25.6248	1.21201
$448$	0	0
$449$	8.36628	0.394829	0.197415	−	0.980320i	$-0.436745\pi$
$449$	8.36628	0.394829	0.197415	+	0.980320i	$0.436745\pi$
$450$	0	0
$451$	−0.945031	−0.0444998
$452$	0	0
$453$	−30.7198	−1.44334
$454$	0	0
$455$	40.6810	1.90715
$456$	0	0
$457$	−35.0653	−1.64028	−0.820142	−	0.572160i	$-0.806105\pi$
$457$	−35.0653	−1.64028	−0.820142	+	0.572160i	$0.806105\pi$
$458$	0	0
$459$	2.86458	0.133707
$460$	0	0
$461$	−31.6273	−1.47303	−0.736515	−	0.676422i	$-0.763530\pi$
$461$	−31.6273	−1.47303	−0.736515	+	0.676422i	$0.763530\pi$
$462$	0	0
$463$	−20.9725	−0.974676	−0.487338	−	0.873213i	$-0.662032\pi$
$463$	−20.9725	−0.974676	−0.487338	+	0.873213i	$0.662032\pi$
$464$	0	0
$465$	72.2422	3.35015
$466$	0	0
$467$	−1.13207	−0.0523858	−0.0261929	−	0.999657i	$-0.508338\pi$
$467$	−1.13207	−0.0523858	−0.0261929	+	0.999657i	$0.508338\pi$
$468$	0	0
$469$	42.0536	1.94186
$470$	0	0
$471$	32.4236	1.49400
$472$	0	0
$473$	4.87904	0.224338
$474$	0	0
$475$	99.8702	4.58236
$476$	0	0
$477$	−2.24507	−0.102795
$478$	0	0
$479$	27.3838	1.25120	0.625599	−	0.780145i	$-0.284855\pi$
$479$	27.3838	1.25120	0.625599	+	0.780145i	$0.284855\pi$
$480$	0	0
$481$	−10.0743	−0.459349
$482$	0	0
$483$	33.4341	1.52130
$484$	0	0
$485$	−41.1965	−1.87063
$486$	0	0
$487$	13.0807	0.592743	0.296371	−	0.955073i	$-0.404223\pi$
$487$	13.0807	0.592743	0.296371	+	0.955073i	$0.404223\pi$
$488$	0	0
$489$	3.75791	0.169939
$490$	0	0
$491$	22.1876	1.00131	0.500655	−	0.865647i	$-0.333092\pi$
$491$	22.1876	1.00131	0.500655	+	0.865647i	$0.333092\pi$
$492$	0	0
$493$	−1.15509	−0.0520228
$494$	0	0
$495$	7.39935	0.332576
$496$	0	0
$497$	−25.6449	−1.15033
$498$	0	0
$499$	−10.8333	−0.484965	−0.242482	−	0.970156i	$-0.577962\pi$
$499$	−10.8333	−0.484965	−0.242482	+	0.970156i	$0.577962\pi$
$500$	0	0
$501$	7.46699	0.333600
$502$	0	0
$503$	−8.58551	−0.382809	−0.191404	−	0.981511i	$-0.561304\pi$
$503$	−8.58551	−0.382809	−0.191404	+	0.981511i	$0.561304\pi$
$504$	0	0
$505$	−35.6559	−1.58667
$506$	0	0
$507$	19.6036	0.870628
$508$	0	0
$509$	12.4973	0.553935	0.276967	−	0.960879i	$-0.410671\pi$
$509$	12.4973	0.553935	0.276967	+	0.960879i	$0.410671\pi$
$510$	0	0
$511$	62.8951	2.78231
$512$	0	0
$513$	28.7115	1.26765
$514$	0	0
$515$	−11.6795	−0.514662
$516$	0	0
$517$	12.9431	0.569236
$518$	0	0
$519$	16.9210	0.742750
$520$	0	0
$521$	−32.1091	−1.40673	−0.703363	−	0.710831i	$-0.748319\pi$
$521$	−32.1091	−1.40673	−0.703363	+	0.710831i	$0.748319\pi$
$522$	0	0
$523$	−31.2693	−1.36731	−0.683655	−	0.729806i	$-0.739611\pi$
$523$	−31.2693	−1.36731	−0.683655	+	0.729806i	$0.739611\pi$
$524$	0	0
$525$	−133.948	−5.84596
$526$	0	0
$527$	−6.68029	−0.290998
$528$	0	0
$529$	−13.2907	−0.577855
$530$	0	0
$531$	−12.4256	−0.539224
$532$	0	0
$533$	−1.26098	−0.0546191
$534$	0	0
$535$	−38.9309	−1.68313
$536$	0	0
$537$	40.3690	1.74205
$538$	0	0
$539$	28.0794	1.20947
$540$	0	0
$541$	20.7911	0.893880	0.446940	−	0.894564i	$-0.352514\pi$
$541$	20.7911	0.893880	0.446940	+	0.894564i	$0.352514\pi$
$542$	0	0
$543$	−4.51423	−0.193724
$544$	0	0
$545$	−31.8417	−1.36395
$546$	0	0
$547$	−24.4070	−1.04357	−0.521785	−	0.853077i	$-0.674734\pi$
$547$	−24.4070	−1.04357	−0.521785	+	0.853077i	$0.674734\pi$
$548$	0	0
$549$	−3.84665	−0.164171
$550$	0	0
$551$	−11.5775	−0.493217
$552$	0	0
$553$	54.1944	2.30458
$554$	0	0
$555$	46.4569	1.97199
$556$	0	0
$557$	−25.3866	−1.07566	−0.537832	−	0.843052i	$-0.680756\pi$
$557$	−25.3866	−1.07566	−0.537832	+	0.843052i	$0.680756\pi$
$558$	0	0
$559$	6.51023	0.275353
$560$	0	0
$561$	−2.31142	−0.0975882
$562$	0	0
$563$	37.9457	1.59922	0.799610	−	0.600520i	$-0.205040\pi$
$563$	37.9457	1.59922	0.799610	+	0.600520i	$0.205040\pi$
$564$	0	0
$565$	−13.1520	−0.553311
$566$	0	0
$567$	−58.1789	−2.44329
$568$	0	0
$569$	−32.9126	−1.37977	−0.689885	−	0.723919i	$-0.742339\pi$
$569$	−32.9126	−1.37977	−0.689885	+	0.723919i	$0.742339\pi$
$570$	0	0
$571$	−34.1309	−1.42834	−0.714168	−	0.699975i	$-0.753195\pi$
$571$	−34.1309	−1.42834	−0.714168	+	0.699975i	$0.753195\pi$
$572$	0	0
$573$	−42.6494	−1.78171
$574$	0	0
$575$	−38.8987	−1.62219
$576$	0	0
$577$	10.1790	0.423758	0.211879	−	0.977296i	$-0.432042\pi$
$577$	10.1790	0.423758	0.211879	+	0.977296i	$0.432042\pi$
$578$	0	0
$579$	23.6331	0.982156
$580$	0	0
$581$	50.8586	2.10997
$582$	0	0
$583$	−2.49661	−0.103399
$584$	0	0
$585$	9.87314	0.408204
$586$	0	0
$587$	−3.38307	−0.139634	−0.0698170	−	0.997560i	$-0.522242\pi$
$587$	−3.38307	−0.139634	−0.0698170	+	0.997560i	$0.522242\pi$
$588$	0	0
$589$	−66.9563	−2.75889
$590$	0	0
$591$	47.5138	1.95446
$592$	0	0
$593$	33.9345	1.39352	0.696761	−	0.717303i	$-0.254624\pi$
$593$	33.9345	1.39352	0.696761	+	0.717303i	$0.254624\pi$
$594$	0	0
$595$	17.3472	0.711167
$596$	0	0
$597$	−7.24080	−0.296346
$598$	0	0
$599$	27.8378	1.13742	0.568710	−	0.822538i	$-0.307443\pi$
$599$	27.8378	1.13742	0.568710	+	0.822538i	$0.307443\pi$
$600$	0	0
$601$	−33.1174	−1.35089	−0.675444	−	0.737411i	$-0.736048\pi$
$601$	−33.1174	−1.35089	−0.675444	+	0.737411i	$0.736048\pi$
$602$	0	0
$603$	10.2063	0.415632
$604$	0	0
$605$	−37.7664	−1.53542
$606$	0	0
$607$	15.0252	0.609853	0.304927	−	0.952376i	$-0.401368\pi$
$607$	15.0252	0.609853	0.304927	+	0.952376i	$0.401368\pi$
$608$	0	0
$609$	15.5279	0.629222
$610$	0	0
$611$	17.2703	0.698681
$612$	0	0
$613$	31.1369	1.25761	0.628803	−	0.777564i	$-0.283545\pi$
$613$	31.1369	1.25761	0.628803	+	0.777564i	$0.283545\pi$
$614$	0	0
$615$	5.81490	0.234480
$616$	0	0
$617$	30.6583	1.23426	0.617129	−	0.786862i	$-0.288296\pi$
$617$	30.6583	1.23426	0.617129	+	0.786862i	$0.288296\pi$
$618$	0	0
$619$	−16.9963	−0.683139	−0.341570	−	0.939856i	$-0.610959\pi$
$619$	−16.9963	−0.683139	−0.341570	+	0.939856i	$0.610959\pi$
$620$	0	0
$621$	−11.1829	−0.448755
$622$	0	0
$623$	16.1618	0.647510
$624$	0	0
$625$	68.4224	2.73690
$626$	0	0
$627$	−23.1673	−0.925211
$628$	0	0
$629$	−4.29590	−0.171289
$630$	0	0
$631$	−42.2222	−1.68084	−0.840420	−	0.541936i	$-0.817691\pi$
$631$	−42.2222	−1.68084	−0.840420	+	0.541936i	$0.817691\pi$
$632$	0	0
$633$	−29.4029	−1.16866
$634$	0	0
$635$	−33.9684	−1.34800
$636$	0	0
$637$	37.4671	1.48450
$638$	0	0
$639$	−6.22394	−0.246215
$640$	0	0
$641$	−26.9584	−1.06479	−0.532396	−	0.846495i	$-0.678708\pi$
$641$	−26.9584	−1.06479	−0.532396	+	0.846495i	$0.678708\pi$
$642$	0	0
$643$	29.1334	1.14891	0.574455	−	0.818536i	$-0.305214\pi$
$643$	29.1334	1.14891	0.574455	+	0.818536i	$0.305214\pi$
$644$	0	0
$645$	−30.0214	−1.18209
$646$	0	0
$647$	40.8634	1.60651	0.803253	−	0.595639i	$-0.203101\pi$
$647$	40.8634	1.60651	0.803253	+	0.595639i	$0.203101\pi$
$648$	0	0
$649$	−13.8177	−0.542393
$650$	0	0
$651$	89.8030	3.51966
$652$	0	0
$653$	45.1628	1.76736	0.883678	−	0.468096i	$-0.155060\pi$
$653$	45.1628	1.76736	0.883678	+	0.468096i	$0.155060\pi$
$654$	0	0
$655$	−50.9563	−1.99103
$656$	0	0
$657$	15.2644	0.595522
$658$	0	0
$659$	38.1992	1.48803	0.744015	−	0.668163i	$-0.232919\pi$
$659$	38.1992	1.48803	0.744015	+	0.668163i	$0.232919\pi$
$660$	0	0
$661$	11.6742	0.454075	0.227038	−	0.973886i	$-0.427096\pi$
$661$	11.6742	0.454075	0.227038	+	0.973886i	$0.427096\pi$
$662$	0	0
$663$	−3.08419	−0.119780
$664$	0	0
$665$	173.871	6.74242
$666$	0	0
$667$	4.50933	0.174602
$668$	0	0
$669$	−5.84218	−0.225872
$670$	0	0
$671$	−4.27762	−0.165136
$672$	0	0
$673$	−31.4825	−1.21356	−0.606781	−	0.794869i	$-0.707540\pi$
$673$	−31.4825	−1.21356	−0.606781	+	0.794869i	$0.707540\pi$
$674$	0	0
$675$	44.8024	1.72444
$676$	0	0
$677$	20.4379	0.785494	0.392747	−	0.919647i	$-0.371525\pi$
$677$	20.4379	0.785494	0.392747	+	0.919647i	$0.371525\pi$
$678$	0	0
$679$	−51.2105	−1.96528
$680$	0	0
$681$	−29.2292	−1.12007
$682$	0	0
$683$	37.2186	1.42413	0.712065	−	0.702113i	$-0.247760\pi$
$683$	37.2186	1.42413	0.712065	+	0.702113i	$0.247760\pi$
$684$	0	0
$685$	−50.8496	−1.94286
$686$	0	0
$687$	−1.02334	−0.0390430
$688$	0	0
$689$	−3.33129	−0.126912
$690$	0	0
$691$	3.15250	0.119927	0.0599633	−	0.998201i	$-0.480902\pi$
$691$	3.15250	0.119927	0.0599633	+	0.998201i	$0.480902\pi$
$692$	0	0
$693$	9.19799	0.349403
$694$	0	0
$695$	−4.74297	−0.179911
$696$	0	0
$697$	−0.537708	−0.0203672
$698$	0	0
$699$	−14.0877	−0.532844
$700$	0	0
$701$	8.26142	0.312030	0.156015	−	0.987755i	$-0.450135\pi$
$701$	8.26142	0.312030	0.156015	+	0.987755i	$0.450135\pi$
$702$	0	0
$703$	−43.0577	−1.62395
$704$	0	0
$705$	−79.6405	−2.99944
$706$	0	0
$707$	−44.3232	−1.66695
$708$	0	0
$709$	39.5326	1.48468	0.742340	−	0.670024i	$-0.233716\pi$
$709$	39.5326	1.48468	0.742340	+	0.670024i	$0.233716\pi$
$710$	0	0
$711$	13.1528	0.493269
$712$	0	0
$713$	26.0790	0.976665
$714$	0	0
$715$	10.9793	0.410603
$716$	0	0
$717$	43.6530	1.63025
$718$	0	0
$719$	14.7694	0.550806	0.275403	−	0.961329i	$-0.411189\pi$
$719$	14.7694	0.550806	0.275403	+	0.961329i	$0.411189\pi$
$720$	0	0
$721$	−14.5186	−0.540702
$722$	0	0
$723$	14.0888	0.523968
$724$	0	0
$725$	−18.0658	−0.670948
$726$	0	0
$727$	38.2081	1.41706	0.708531	−	0.705680i	$-0.249358\pi$
$727$	38.2081	1.41706	0.708531	+	0.705680i	$0.249358\pi$
$728$	0	0
$729$	8.10620	0.300230
$730$	0	0
$731$	2.77610	0.102678
$732$	0	0
$733$	−11.1059	−0.410204	−0.205102	−	0.978741i	$-0.565753\pi$
$733$	−11.1059	−0.410204	−0.205102	+	0.978741i	$0.565753\pi$
$734$	0	0
$735$	−172.777	−6.37296
$736$	0	0
$737$	11.3498	0.418074
$738$	0	0
$739$	−8.58551	−0.315823	−0.157912	−	0.987453i	$-0.550476\pi$
$739$	−8.58551	−0.315823	−0.157912	+	0.987453i	$0.550476\pi$
$740$	0	0
$741$	−30.9127	−1.13561
$742$	0	0
$743$	−24.6822	−0.905502	−0.452751	−	0.891637i	$-0.649557\pi$
$743$	−24.6822	−0.905502	−0.452751	+	0.891637i	$0.649557\pi$
$744$	0	0
$745$	−51.9035	−1.90160
$746$	0	0
$747$	12.3432	0.451614
$748$	0	0
$749$	−48.3943	−1.76829
$750$	0	0
$751$	37.5795	1.37129	0.685647	−	0.727934i	$-0.259519\pi$
$751$	37.5795	1.37129	0.685647	+	0.727934i	$0.259519\pi$
$752$	0	0
$753$	−17.3853	−0.633555
$754$	0	0
$755$	62.2235	2.26454
$756$	0	0
$757$	0.860873	0.0312890	0.0156445	−	0.999878i	$-0.495020\pi$
$757$	0.860873	0.0312890	0.0156445	+	0.999878i	$0.495020\pi$
$758$	0	0
$759$	9.02347	0.327531
$760$	0	0
$761$	38.4549	1.39399	0.696994	−	0.717077i	$-0.254521\pi$
$761$	38.4549	1.39399	0.696994	+	0.717077i	$0.254521\pi$
$762$	0	0
$763$	−39.5819	−1.43296
$764$	0	0
$765$	4.21012	0.152217
$766$	0	0
$767$	−18.4374	−0.665734
$768$	0	0
$769$	−30.0822	−1.08479	−0.542396	−	0.840123i	$-0.682483\pi$
$769$	−30.0822	−1.08479	−0.542396	+	0.840123i	$0.682483\pi$
$770$	0	0
$771$	−51.7090	−1.86225
$772$	0	0
$773$	−0.282341	−0.0101551	−0.00507755	−	0.999987i	$-0.501616\pi$
$773$	−0.282341	−0.0101551	−0.00507755	+	0.999987i	$0.501616\pi$
$774$	0	0
$775$	−104.481	−3.75305
$776$	0	0
$777$	57.7497	2.07176
$778$	0	0
$779$	−5.38943	−0.193096
$780$	0	0
$781$	−6.92127	−0.247662
$782$	0	0
$783$	−5.19372	−0.185608
$784$	0	0
$785$	−65.6745	−2.34402
$786$	0	0
$787$	5.76391	0.205461	0.102731	−	0.994709i	$-0.467242\pi$
$787$	5.76391	0.205461	0.102731	+	0.994709i	$0.467242\pi$
$788$	0	0
$789$	47.7635	1.70043
$790$	0	0
$791$	−16.3491	−0.581306
$792$	0	0
$793$	−5.70774	−0.202688
$794$	0	0
$795$	15.3620	0.544833
$796$	0	0
$797$	31.3364	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$797$	31.3364	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$798$	0	0
$799$	7.36442	0.260534
$800$	0	0
$801$	3.92242	0.138592
$802$	0	0
$803$	16.9746	0.599022
$804$	0	0
$805$	−67.7213	−2.38686
$806$	0	0
$807$	46.3180	1.63047
$808$	0	0
$809$	−17.1022	−0.601280	−0.300640	−	0.953738i	$-0.597200\pi$
$809$	−17.1022	−0.601280	−0.300640	+	0.953738i	$0.597200\pi$
$810$	0	0
$811$	−20.5584	−0.721904	−0.360952	−	0.932584i	$-0.617548\pi$
$811$	−20.5584	−0.721904	−0.360952	+	0.932584i	$0.617548\pi$
$812$	0	0
$813$	−54.1686	−1.89978
$814$	0	0
$815$	−7.61171	−0.266627
$816$	0	0
$817$	27.8247	0.973465
$818$	0	0
$819$	12.2731	0.428857
$820$	0	0
$821$	−24.6047	−0.858710	−0.429355	−	0.903136i	$-0.641259\pi$
$821$	−24.6047	−0.858710	−0.429355	+	0.903136i	$0.641259\pi$
$822$	0	0
$823$	−42.0754	−1.46666	−0.733328	−	0.679875i	$-0.762034\pi$
$823$	−42.0754	−1.46666	−0.733328	+	0.679875i	$0.762034\pi$
$824$	0	0
$825$	−36.1509	−1.25861
$826$	0	0
$827$	−35.2966	−1.22738	−0.613692	−	0.789546i	$-0.710316\pi$
$827$	−35.2966	−1.22738	−0.613692	+	0.789546i	$0.710316\pi$
$828$	0	0
$829$	14.2598	0.495263	0.247632	−	0.968854i	$-0.420348\pi$
$829$	14.2598	0.495263	0.247632	+	0.968854i	$0.420348\pi$
$830$	0	0
$831$	46.5565	1.61503
$832$	0	0
$833$	15.9768	0.553563
$834$	0	0
$835$	−15.1245	−0.523405
$836$	0	0
$837$	−30.0370	−1.03823
$838$	0	0
$839$	−40.1648	−1.38664	−0.693321	−	0.720629i	$-0.743853\pi$
$839$	−40.1648	−1.38664	−0.693321	+	0.720629i	$0.743853\pi$
$840$	0	0
$841$	−26.9057	−0.927783
$842$	0	0
$843$	15.0398	0.518000
$844$	0	0
$845$	−39.7074	−1.36598
$846$	0	0
$847$	−46.9467	−1.61311
$848$	0	0
$849$	−26.5229	−0.910264
$850$	0	0
$851$	16.7706	0.574890
$852$	0	0
$853$	12.8913	0.441390	0.220695	−	0.975343i	$-0.429167\pi$
$853$	12.8913	0.441390	0.220695	+	0.975343i	$0.429167\pi$
$854$	0	0
$855$	42.1978	1.44314
$856$	0	0
$857$	−30.2255	−1.03248	−0.516242	−	0.856443i	$-0.672670\pi$
$857$	−30.2255	−1.03248	−0.516242	+	0.856443i	$0.672670\pi$
$858$	0	0
$859$	47.3821	1.61665	0.808327	−	0.588733i	$-0.200373\pi$
$859$	47.3821	1.61665	0.808327	+	0.588733i	$0.200373\pi$
$860$	0	0
$861$	7.22840	0.246343
$862$	0	0
$863$	−10.5243	−0.358250	−0.179125	−	0.983826i	$-0.557327\pi$
$863$	−10.5243	−0.358250	−0.179125	+	0.983826i	$0.557327\pi$
$864$	0	0
$865$	−34.2738	−1.16534
$866$	0	0
$867$	33.7785	1.14718
$868$	0	0
$869$	14.6264	0.496168
$870$	0	0
$871$	15.1443	0.513145
$872$	0	0
$873$	−12.4286	−0.420645
$874$	0	0
$875$	162.645	5.49841
$876$	0	0
$877$	−28.9234	−0.976674	−0.488337	−	0.872655i	$-0.662396\pi$
$877$	−28.9234	−0.976674	−0.488337	+	0.872655i	$0.662396\pi$
$878$	0	0
$879$	−55.1481	−1.86010
$880$	0	0
$881$	−33.9904	−1.14517	−0.572583	−	0.819847i	$-0.694059\pi$
$881$	−33.9904	−1.14517	−0.572583	+	0.819847i	$0.694059\pi$
$882$	0	0
$883$	3.88401	0.130707	0.0653537	−	0.997862i	$-0.479182\pi$
$883$	3.88401	0.130707	0.0653537	+	0.997862i	$0.479182\pi$
$884$	0	0
$885$	85.0224	2.85800
$886$	0	0
$887$	−22.7330	−0.763298	−0.381649	−	0.924307i	$-0.624644\pi$
$887$	−22.7330	−0.763298	−0.381649	+	0.924307i	$0.624644\pi$
$888$	0	0
$889$	−42.2255	−1.41620
$890$	0	0
$891$	−15.7018	−0.526030
$892$	0	0
$893$	73.8133	2.47007
$894$	0	0
$895$	−81.7681	−2.73320
$896$	0	0
$897$	12.0402	0.402012
$898$	0	0
$899$	12.1119	0.403955
$900$	0	0
$901$	−1.42053	−0.0473248
$902$	0	0
$903$	−37.3190	−1.24190
$904$	0	0
$905$	9.14364	0.303945
$906$	0	0
$907$	−11.2618	−0.373941	−0.186970	−	0.982366i	$-0.559867\pi$
$907$	−11.2618	−0.373941	−0.186970	+	0.982366i	$0.559867\pi$
$908$	0	0
$909$	−10.7571	−0.356790
$910$	0	0
$911$	−44.6942	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$911$	−44.6942	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$912$	0	0
$913$	13.7261	0.454268
$914$	0	0
$915$	26.3208	0.870139
$916$	0	0
$917$	−63.3428	−2.09176
$918$	0	0
$919$	34.2776	1.13071	0.565356	−	0.824847i	$-0.308739\pi$
$919$	34.2776	1.13071	0.565356	+	0.824847i	$0.308739\pi$
$920$	0	0
$921$	0.435314	0.0143441
$922$	0	0
$923$	−9.23522	−0.303981
$924$	0	0
$925$	−67.1885	−2.20914
$926$	0	0
$927$	−3.52362	−0.115731
$928$	0	0
$929$	−13.1894	−0.432730	−0.216365	−	0.976313i	$-0.569420\pi$
$929$	−13.1894	−0.432730	−0.216365	+	0.976313i	$0.569420\pi$
$930$	0	0
$931$	160.135	5.24820
$932$	0	0
$933$	30.7375	1.00630
$934$	0	0
$935$	4.68181	0.153112
$936$	0	0
$937$	25.0892	0.819628	0.409814	−	0.912169i	$-0.365594\pi$
$937$	25.0892	0.819628	0.409814	+	0.912169i	$0.365594\pi$
$938$	0	0
$939$	57.2353	1.86780
$940$	0	0
$941$	33.9061	1.10531	0.552653	−	0.833411i	$-0.313615\pi$
$941$	33.9061	1.10531	0.552653	+	0.833411i	$0.313615\pi$
$942$	0	0
$943$	2.09914	0.0683574
$944$	0	0
$945$	77.9994	2.53732
$946$	0	0
$947$	−18.9560	−0.615989	−0.307994	−	0.951388i	$-0.599658\pi$
$947$	−18.9560	−0.615989	−0.307994	+	0.951388i	$0.599658\pi$
$948$	0	0
$949$	22.6497	0.735240
$950$	0	0
$951$	34.6364	1.12316
$952$	0	0
$953$	−10.6442	−0.344798	−0.172399	−	0.985027i	$-0.555152\pi$
$953$	−10.6442	−0.344798	−0.172399	+	0.985027i	$0.555152\pi$
$954$	0	0
$955$	86.3870	2.79542
$956$	0	0
$957$	4.19080	0.135469
$958$	0	0
$959$	−63.2102	−2.04116
$960$	0	0
$961$	39.0473	1.25959
$962$	0	0
$963$	−11.7451	−0.378482
$964$	0	0
$965$	−47.8691	−1.54096
$966$	0	0
$967$	−8.97759	−0.288700	−0.144350	−	0.989527i	$-0.546109\pi$
$967$	−8.97759	−0.288700	−0.144350	+	0.989527i	$0.546109\pi$
$968$	0	0
$969$	−13.1818	−0.423461
$970$	0	0
$971$	28.6301	0.918785	0.459392	−	0.888233i	$-0.348067\pi$
$971$	28.6301	0.918785	0.459392	+	0.888233i	$0.348067\pi$
$972$	0	0
$973$	−5.89590	−0.189014
$974$	0	0
$975$	−48.2371	−1.54482
$976$	0	0
$977$	−55.6811	−1.78140	−0.890699	−	0.454594i	$-0.849784\pi$
$977$	−55.6811	−1.78140	−0.890699	+	0.454594i	$0.849784\pi$
$978$	0	0
$979$	4.36189	0.139407
$980$	0	0
$981$	−9.60639	−0.306708
$982$	0	0
$983$	12.2311	0.390112	0.195056	−	0.980792i	$-0.437511\pi$
$983$	12.2311	0.390112	0.195056	+	0.980792i	$0.437511\pi$
$984$	0	0
$985$	−96.2400	−3.06646
$986$	0	0
$987$	−98.9997	−3.15119
$988$	0	0
$989$	−10.8375	−0.344613
$990$	0	0
$991$	−0.482440	−0.0153252	−0.00766260	−	0.999971i	$-0.502439\pi$
$991$	−0.482440	−0.0153252	−0.00766260	+	0.999971i	$0.502439\pi$
$992$	0	0
$993$	−18.1262	−0.575218
$994$	0	0
$995$	14.6663	0.464955
$996$	0	0
$997$	36.6265	1.15997	0.579987	−	0.814626i	$-0.303058\pi$
$997$	36.6265	1.15997	0.579987	+	0.814626i	$0.303058\pi$
$998$	0	0
$999$	−19.3159	−0.611129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.15	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.15	✓	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)

\(f(q)\)	\(=\)	\(q-2.06433 q^{3} +4.18134 q^{5} +5.19775 q^{7} +1.26148 q^{9} +O(q^{10})\)	\(q-2.06433 q^{3} +4.18134 q^{5} +5.19775 q^{7} +1.26148 q^{9} +1.40281 q^{11} +1.87181 q^{13} -8.63168 q^{15} +0.798178 q^{17} +8.00011 q^{19} -10.7299 q^{21} -3.11598 q^{23} +12.4836 q^{25} +3.58889 q^{27} -1.44716 q^{29} -8.36943 q^{31} -2.89587 q^{33} +21.7335 q^{35} -5.38214 q^{37} -3.86403 q^{39} -0.673670 q^{41} +3.47805 q^{43} +5.27466 q^{45} +9.22654 q^{47} +20.0166 q^{49} -1.64771 q^{51} -1.77972 q^{53} +5.86563 q^{55} -16.5149 q^{57} -9.85004 q^{59} -3.04932 q^{61} +6.55683 q^{63} +7.82666 q^{65} +8.09074 q^{67} +6.43242 q^{69} -4.93386 q^{71} +12.1005 q^{73} -25.7703 q^{75} +7.29145 q^{77} +10.4265 q^{79} -11.1931 q^{81} +9.78473 q^{83} +3.33745 q^{85} +2.98743 q^{87} +3.10939 q^{89} +9.72917 q^{91} +17.2773 q^{93} +33.4512 q^{95} -9.85245 q^{97} +1.76961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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