LMFDB - Embedded newform 6044.2.a.b.1.19

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			1.19
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)

$f(q)$	$=$	$q-1.08193 q^{3} -0.304592 q^{5} -4.98661 q^{7} -1.82943 q^{9} +O(q^{10})$	$q-1.08193 q^{3} -0.304592 q^{5} -4.98661 q^{7} -1.82943 q^{9} +3.37038 q^{11} +3.20513 q^{13} +0.329546 q^{15} -5.45302 q^{17} +3.52218 q^{19} +5.39515 q^{21} -7.18904 q^{23} -4.90722 q^{25} +5.22509 q^{27} +2.29133 q^{29} -1.32910 q^{31} -3.64650 q^{33} +1.51888 q^{35} -10.4595 q^{37} -3.46771 q^{39} +3.05671 q^{41} -10.3383 q^{43} +0.557231 q^{45} +4.97175 q^{47} +17.8663 q^{49} +5.89977 q^{51} -1.36423 q^{53} -1.02659 q^{55} -3.81074 q^{57} -5.99176 q^{59} -8.24929 q^{61} +9.12268 q^{63} -0.976256 q^{65} +2.00122 q^{67} +7.77802 q^{69} -12.2356 q^{71} -8.86889 q^{73} +5.30926 q^{75} -16.8068 q^{77} +14.9659 q^{79} -0.164863 q^{81} +8.59988 q^{83} +1.66095 q^{85} -2.47905 q^{87} -7.40042 q^{89} -15.9827 q^{91} +1.43799 q^{93} -1.07283 q^{95} +1.75421 q^{97} -6.16589 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.08193	−0.624651	−0.312325	−	0.949975i	$-0.601108\pi$
$3$	−1.08193	−0.624651	−0.312325	+	0.949975i	$0.601108\pi$
$4$	0	0
$5$	−0.304592	−0.136218	−0.0681088	−	0.997678i	$-0.521697\pi$
$5$	−0.304592	−0.136218	−0.0681088	+	0.997678i	$0.521697\pi$
$6$	0	0
$7$	−4.98661	−1.88476	−0.942381	−	0.334542i	$-0.891418\pi$
$7$	−4.98661	−1.88476	−0.942381	+	0.334542i	$0.891418\pi$
$8$	0	0
$9$	−1.82943	−0.609812
$10$	0	0
$11$	3.37038	1.01621	0.508104	−	0.861296i	$-0.330347\pi$
$11$	3.37038	1.01621	0.508104	+	0.861296i	$0.330347\pi$
$12$	0	0
$13$	3.20513	0.888942	0.444471	−	0.895793i	$-0.353392\pi$
$13$	3.20513	0.888942	0.444471	+	0.895793i	$0.353392\pi$
$14$	0	0
$15$	0.329546	0.0850884
$16$	0	0
$17$	−5.45302	−1.32255	−0.661276	−	0.750142i	$-0.729985\pi$
$17$	−5.45302	−1.32255	−0.661276	+	0.750142i	$0.729985\pi$
$18$	0	0
$19$	3.52218	0.808044	0.404022	−	0.914749i	$-0.367612\pi$
$19$	3.52218	0.808044	0.404022	+	0.914749i	$0.367612\pi$
$20$	0	0
$21$	5.39515	1.17732
$22$	0	0
$23$	−7.18904	−1.49902	−0.749510	−	0.661993i	$-0.769711\pi$
$23$	−7.18904	−1.49902	−0.749510	+	0.661993i	$0.769711\pi$
$24$	0	0
$25$	−4.90722	−0.981445
$26$	0	0
$27$	5.22509	1.00557
$28$	0	0
$29$	2.29133	0.425490	0.212745	−	0.977108i	$-0.431760\pi$
$29$	2.29133	0.425490	0.212745	+	0.977108i	$0.431760\pi$
$30$	0	0
$31$	−1.32910	−0.238714	−0.119357	−	0.992851i	$-0.538083\pi$
$31$	−1.32910	−0.238714	−0.119357	+	0.992851i	$0.538083\pi$
$32$	0	0
$33$	−3.64650	−0.634775
$34$	0	0
$35$	1.51888	0.256738
$36$	0	0
$37$	−10.4595	−1.71953	−0.859766	−	0.510688i	$-0.829391\pi$
$37$	−10.4595	−1.71953	−0.859766	+	0.510688i	$0.829391\pi$
$38$	0	0
$39$	−3.46771	−0.555278
$40$	0	0
$41$	3.05671	0.477378	0.238689	−	0.971096i	$-0.423282\pi$
$41$	3.05671	0.477378	0.238689	+	0.971096i	$0.423282\pi$
$42$	0	0
$43$	−10.3383	−1.57657	−0.788285	−	0.615310i	$-0.789031\pi$
$43$	−10.3383	−1.57657	−0.788285	+	0.615310i	$0.789031\pi$
$44$	0	0
$45$	0.557231	0.0830671
$46$	0	0
$47$	4.97175	0.725204	0.362602	−	0.931944i	$-0.381888\pi$
$47$	4.97175	0.725204	0.362602	+	0.931944i	$0.381888\pi$
$48$	0	0
$49$	17.8663	2.55233
$50$	0	0
$51$	5.89977	0.826133
$52$	0	0
$53$	−1.36423	−0.187392	−0.0936959	−	0.995601i	$-0.529868\pi$
$53$	−1.36423	−0.187392	−0.0936959	+	0.995601i	$0.529868\pi$
$54$	0	0
$55$	−1.02659	−0.138425
$56$	0	0
$57$	−3.81074	−0.504745
$58$	0	0
$59$	−5.99176	−0.780061	−0.390031	−	0.920802i	$-0.627536\pi$
$59$	−5.99176	−0.780061	−0.390031	+	0.920802i	$0.627536\pi$
$60$	0	0
$61$	−8.24929	−1.05621	−0.528107	−	0.849178i	$-0.677098\pi$
$61$	−8.24929	−1.05621	−0.528107	+	0.849178i	$0.677098\pi$
$62$	0	0
$63$	9.12268	1.14935
$64$	0	0
$65$	−0.976256	−0.121090
$66$	0	0
$67$	2.00122	0.244488	0.122244	−	0.992500i	$-0.460991\pi$
$67$	2.00122	0.244488	0.122244	+	0.992500i	$0.460991\pi$
$68$	0	0
$69$	7.77802	0.936363
$70$	0	0
$71$	−12.2356	−1.45210	−0.726051	−	0.687641i	$-0.758646\pi$
$71$	−12.2356	−1.45210	−0.726051	+	0.687641i	$0.758646\pi$
$72$	0	0
$73$	−8.86889	−1.03803	−0.519013	−	0.854767i	$-0.673700\pi$
$73$	−8.86889	−1.03803	−0.519013	+	0.854767i	$0.673700\pi$
$74$	0	0
$75$	5.30926	0.613060
$76$	0	0
$77$	−16.8068	−1.91531
$78$	0	0
$79$	14.9659	1.68379	0.841896	−	0.539640i	$-0.181440\pi$
$79$	14.9659	1.68379	0.841896	+	0.539640i	$0.181440\pi$
$80$	0	0
$81$	−0.164863	−0.0183181
$82$	0	0
$83$	8.59988	0.943959	0.471980	−	0.881609i	$-0.343540\pi$
$83$	8.59988	0.943959	0.471980	+	0.881609i	$0.343540\pi$
$84$	0	0
$85$	1.66095	0.180155
$86$	0	0
$87$	−2.47905	−0.265783
$88$	0	0
$89$	−7.40042	−0.784443	−0.392222	−	0.919871i	$-0.628293\pi$
$89$	−7.40042	−0.784443	−0.392222	+	0.919871i	$0.628293\pi$
$90$	0	0
$91$	−15.9827	−1.67544
$92$	0	0
$93$	1.43799	0.149113
$94$	0	0
$95$	−1.07283	−0.110070
$96$	0	0
$97$	1.75421	0.178113	0.0890565	−	0.996027i	$-0.471615\pi$
$97$	1.75421	0.178113	0.0890565	+	0.996027i	$0.471615\pi$
$98$	0	0
$99$	−6.16589	−0.619696
$100$	0	0
$101$	−3.90726	−0.388787	−0.194393	−	0.980924i	$-0.562274\pi$
$101$	−3.90726	−0.388787	−0.194393	+	0.980924i	$0.562274\pi$
$102$	0	0
$103$	16.6057	1.63621	0.818103	−	0.575071i	$-0.195026\pi$
$103$	16.6057	1.63621	0.818103	+	0.575071i	$0.195026\pi$
$104$	0	0
$105$	−1.64332	−0.160371
$106$	0	0
$107$	−11.0183	−1.06518	−0.532590	−	0.846374i	$-0.678781\pi$
$107$	−11.0183	−1.06518	−0.532590	+	0.846374i	$0.678781\pi$
$108$	0	0
$109$	4.05106	0.388021	0.194011	−	0.980999i	$-0.437850\pi$
$109$	4.05106	0.388021	0.194011	+	0.980999i	$0.437850\pi$
$110$	0	0
$111$	11.3164	1.07411
$112$	0	0
$113$	−4.31765	−0.406171	−0.203085	−	0.979161i	$-0.565097\pi$
$113$	−4.31765	−0.406171	−0.203085	+	0.979161i	$0.565097\pi$
$114$	0	0
$115$	2.18972	0.204193
$116$	0	0
$117$	−5.86357	−0.542087
$118$	0	0
$119$	27.1921	2.49270
$120$	0	0
$121$	0.359469	0.0326790
$122$	0	0
$123$	−3.30714	−0.298195
$124$	0	0
$125$	3.01766	0.269908
$126$	0	0
$127$	−18.6823	−1.65779	−0.828893	−	0.559408i	$-0.811029\pi$
$127$	−18.6823	−1.65779	−0.828893	+	0.559408i	$0.811029\pi$
$128$	0	0
$129$	11.1852	0.984806
$130$	0	0
$131$	−13.1589	−1.14970	−0.574851	−	0.818258i	$-0.694940\pi$
$131$	−13.1589	−1.14970	−0.574851	+	0.818258i	$0.694940\pi$
$132$	0	0
$133$	−17.5638	−1.52297
$134$	0	0
$135$	−1.59152	−0.136976
$136$	0	0
$137$	15.6606	1.33798	0.668988	−	0.743273i	$-0.266728\pi$
$137$	15.6606	1.33798	0.668988	+	0.743273i	$0.266728\pi$
$138$	0	0
$139$	9.27219	0.786457	0.393228	−	0.919441i	$-0.371358\pi$
$139$	9.27219	0.786457	0.393228	+	0.919441i	$0.371358\pi$
$140$	0	0
$141$	−5.37907	−0.452999
$142$	0	0
$143$	10.8025	0.903350
$144$	0	0
$145$	−0.697922	−0.0579592
$146$	0	0
$147$	−19.3300	−1.59431
$148$	0	0
$149$	11.4337	0.936682	0.468341	−	0.883548i	$-0.344852\pi$
$149$	11.4337	0.936682	0.468341	+	0.883548i	$0.344852\pi$
$150$	0	0
$151$	16.4710	1.34039	0.670195	−	0.742185i	$-0.266211\pi$
$151$	16.4710	1.34039	0.670195	+	0.742185i	$0.266211\pi$
$152$	0	0
$153$	9.97595	0.806508
$154$	0	0
$155$	0.404834	0.0325171
$156$	0	0
$157$	11.5914	0.925095	0.462547	−	0.886595i	$-0.346935\pi$
$157$	11.5914	0.925095	0.462547	+	0.886595i	$0.346935\pi$
$158$	0	0
$159$	1.47600	0.117054
$160$	0	0
$161$	35.8490	2.82529
$162$	0	0
$163$	−10.3582	−0.811314	−0.405657	−	0.914025i	$-0.632957\pi$
$163$	−10.3582	−0.811314	−0.405657	+	0.914025i	$0.632957\pi$
$164$	0	0
$165$	1.11070	0.0864676
$166$	0	0
$167$	13.6189	1.05386	0.526930	−	0.849909i	$-0.323343\pi$
$167$	13.6189	1.05386	0.526930	+	0.849909i	$0.323343\pi$
$168$	0	0
$169$	−2.72717	−0.209782
$170$	0	0
$171$	−6.44360	−0.492755
$172$	0	0
$173$	2.56072	0.194688	0.0973440	−	0.995251i	$-0.468965\pi$
$173$	2.56072	0.194688	0.0973440	+	0.995251i	$0.468965\pi$
$174$	0	0
$175$	24.4704	1.84979
$176$	0	0
$177$	6.48265	0.487266
$178$	0	0
$179$	13.8365	1.03419	0.517093	−	0.855929i	$-0.327014\pi$
$179$	13.8365	1.03419	0.517093	+	0.855929i	$0.327014\pi$
$180$	0	0
$181$	8.61560	0.640393	0.320196	−	0.947351i	$-0.396251\pi$
$181$	8.61560	0.640393	0.320196	+	0.947351i	$0.396251\pi$
$182$	0	0
$183$	8.92513	0.659764
$184$	0	0
$185$	3.18588	0.234231
$186$	0	0
$187$	−18.3788	−1.34399
$188$	0	0
$189$	−26.0555	−1.89526
$190$	0	0
$191$	17.8301	1.29014	0.645072	−	0.764122i	$-0.276827\pi$
$191$	17.8301	1.29014	0.645072	+	0.764122i	$0.276827\pi$
$192$	0	0
$193$	−21.1811	−1.52465	−0.762325	−	0.647195i	$-0.775942\pi$
$193$	−21.1811	−1.52465	−0.762325	+	0.647195i	$0.775942\pi$
$194$	0	0
$195$	1.05624	0.0756387
$196$	0	0
$197$	23.1974	1.65275	0.826373	−	0.563123i	$-0.190400\pi$
$197$	23.1974	1.65275	0.826373	+	0.563123i	$0.190400\pi$
$198$	0	0
$199$	11.8485	0.839915	0.419958	−	0.907544i	$-0.362045\pi$
$199$	11.8485	0.839915	0.419958	+	0.907544i	$0.362045\pi$
$200$	0	0
$201$	−2.16518	−0.152720
$202$	0	0
$203$	−11.4260	−0.801947
$204$	0	0
$205$	−0.931050	−0.0650274
$206$	0	0
$207$	13.1519	0.914119
$208$	0	0
$209$	11.8711	0.821141
$210$	0	0
$211$	16.5026	1.13608	0.568042	−	0.823000i	$-0.307701\pi$
$211$	16.5026	1.13608	0.568042	+	0.823000i	$0.307701\pi$
$212$	0	0
$213$	13.2380	0.907056
$214$	0	0
$215$	3.14895	0.214757
$216$	0	0
$217$	6.62773	0.449919
$218$	0	0
$219$	9.59549	0.648403
$220$	0	0
$221$	−17.4776	−1.17567
$222$	0	0
$223$	4.34325	0.290845	0.145423	−	0.989370i	$-0.453546\pi$
$223$	4.34325	0.290845	0.145423	+	0.989370i	$0.453546\pi$
$224$	0	0
$225$	8.97745	0.598496
$226$	0	0
$227$	24.4607	1.62351	0.811756	−	0.583996i	$-0.198512\pi$
$227$	24.4607	1.62351	0.811756	+	0.583996i	$0.198512\pi$
$228$	0	0
$229$	−19.3190	−1.27663	−0.638317	−	0.769773i	$-0.720369\pi$
$229$	−19.3190	−1.27663	−0.638317	+	0.769773i	$0.720369\pi$
$230$	0	0
$231$	18.1837	1.19640
$232$	0	0
$233$	−22.6355	−1.48290	−0.741452	−	0.671006i	$-0.765862\pi$
$233$	−22.6355	−1.48290	−0.741452	+	0.671006i	$0.765862\pi$
$234$	0	0
$235$	−1.51436	−0.0987856
$236$	0	0
$237$	−16.1920	−1.05178
$238$	0	0
$239$	−20.2648	−1.31082	−0.655409	−	0.755274i	$-0.727504\pi$
$239$	−20.2648	−1.31082	−0.655409	+	0.755274i	$0.727504\pi$
$240$	0	0
$241$	3.28953	0.211897	0.105949	−	0.994372i	$-0.466212\pi$
$241$	3.28953	0.211897	0.105949	+	0.994372i	$0.466212\pi$
$242$	0	0
$243$	−15.4969	−0.994127
$244$	0	0
$245$	−5.44193	−0.347672
$246$	0	0
$247$	11.2890	0.718304
$248$	0	0
$249$	−9.30444	−0.589645
$250$	0	0
$251$	11.5709	0.730348	0.365174	−	0.930939i	$-0.381009\pi$
$251$	11.5709	0.730348	0.365174	+	0.930939i	$0.381009\pi$
$252$	0	0
$253$	−24.2298	−1.52332
$254$	0	0
$255$	−1.79702	−0.112534
$256$	0	0
$257$	−0.0666605	−0.00415817	−0.00207909	−	0.999998i	$-0.500662\pi$
$257$	−0.0666605	−0.00415817	−0.00207909	+	0.999998i	$0.500662\pi$
$258$	0	0
$259$	52.1575	3.24091
$260$	0	0
$261$	−4.19185	−0.259469
$262$	0	0
$263$	23.2067	1.43099	0.715493	−	0.698620i	$-0.246202\pi$
$263$	23.2067	1.43099	0.715493	+	0.698620i	$0.246202\pi$
$264$	0	0
$265$	0.415534	0.0255261
$266$	0	0
$267$	8.00672	0.490003
$268$	0	0
$269$	12.8642	0.784345	0.392172	−	0.919892i	$-0.371724\pi$
$269$	12.8642	0.784345	0.392172	+	0.919892i	$0.371724\pi$
$270$	0	0
$271$	25.1384	1.52705	0.763523	−	0.645781i	$-0.223468\pi$
$271$	25.1384	1.52705	0.763523	+	0.645781i	$0.223468\pi$
$272$	0	0
$273$	17.2921	1.04657
$274$	0	0
$275$	−16.5392	−0.997352
$276$	0	0
$277$	6.44100	0.387002	0.193501	−	0.981100i	$-0.438016\pi$
$277$	6.44100	0.387002	0.193501	+	0.981100i	$0.438016\pi$
$278$	0	0
$279$	2.43151	0.145571
$280$	0	0
$281$	−25.1742	−1.50177	−0.750883	−	0.660436i	$-0.770372\pi$
$281$	−25.1742	−1.50177	−0.750883	+	0.660436i	$0.770372\pi$
$282$	0	0
$283$	−7.50154	−0.445921	−0.222960	−	0.974828i	$-0.571572\pi$
$283$	−7.50154	−0.445921	−0.222960	+	0.974828i	$0.571572\pi$
$284$	0	0
$285$	1.16072	0.0687552
$286$	0	0
$287$	−15.2426	−0.899745
$288$	0	0
$289$	12.7355	0.749146
$290$	0	0
$291$	−1.89793	−0.111258
$292$	0	0
$293$	−30.5952	−1.78739	−0.893694	−	0.448677i	$-0.851896\pi$
$293$	−30.5952	−1.78739	−0.893694	+	0.448677i	$0.851896\pi$
$294$	0	0
$295$	1.82504	0.106258
$296$	0	0
$297$	17.6106	1.02187
$298$	0	0
$299$	−23.0418	−1.33254
$300$	0	0
$301$	51.5529	2.97146
$302$	0	0
$303$	4.22736	0.242856
$304$	0	0
$305$	2.51267	0.143875
$306$	0	0
$307$	13.6002	0.776207	0.388104	−	0.921616i	$-0.373130\pi$
$307$	13.6002	0.776207	0.388104	+	0.921616i	$0.373130\pi$
$308$	0	0
$309$	−17.9661	−1.02206
$310$	0	0
$311$	13.4887	0.764873	0.382437	−	0.923982i	$-0.375085\pi$
$311$	13.4887	0.764873	0.382437	+	0.923982i	$0.375085\pi$
$312$	0	0
$313$	27.1852	1.53660	0.768300	−	0.640090i	$-0.221103\pi$
$313$	27.1852	1.53660	0.768300	+	0.640090i	$0.221103\pi$
$314$	0	0
$315$	−2.77869	−0.156562
$316$	0	0
$317$	7.65742	0.430083	0.215042	−	0.976605i	$-0.431011\pi$
$317$	7.65742	0.430083	0.215042	+	0.976605i	$0.431011\pi$
$318$	0	0
$319$	7.72267	0.432386
$320$	0	0
$321$	11.9210	0.665365
$322$	0	0
$323$	−19.2065	−1.06868
$324$	0	0
$325$	−15.7283	−0.872447
$326$	0	0
$327$	−4.38295	−0.242378
$328$	0	0
$329$	−24.7922	−1.36684
$330$	0	0
$331$	4.69041	0.257808	0.128904	−	0.991657i	$-0.458854\pi$
$331$	4.69041	0.257808	0.128904	+	0.991657i	$0.458854\pi$
$332$	0	0
$333$	19.1350	1.04859
$334$	0	0
$335$	−0.609557	−0.0333036
$336$	0	0
$337$	−7.67848	−0.418273	−0.209137	−	0.977886i	$-0.567065\pi$
$337$	−7.67848	−0.418273	−0.209137	+	0.977886i	$0.567065\pi$
$338$	0	0
$339$	4.67138	0.253715
$340$	0	0
$341$	−4.47959	−0.242583
$342$	0	0
$343$	−54.1860	−2.92577
$344$	0	0
$345$	−2.36912	−0.127549
$346$	0	0
$347$	26.9003	1.44408	0.722042	−	0.691849i	$-0.243204\pi$
$347$	26.9003	1.44408	0.722042	+	0.691849i	$0.243204\pi$
$348$	0	0
$349$	−2.44594	−0.130928	−0.0654641	−	0.997855i	$-0.520853\pi$
$349$	−2.44594	−0.130928	−0.0654641	+	0.997855i	$0.520853\pi$
$350$	0	0
$351$	16.7471	0.893893
$352$	0	0
$353$	25.4153	1.35272	0.676360	−	0.736571i	$-0.263556\pi$
$353$	25.4153	1.35272	0.676360	+	0.736571i	$0.263556\pi$
$354$	0	0
$355$	3.72687	0.197802
$356$	0	0
$357$	−29.4199	−1.55706
$358$	0	0
$359$	26.3248	1.38937	0.694685	−	0.719314i	$-0.255544\pi$
$359$	26.3248	1.38937	0.694685	+	0.719314i	$0.255544\pi$
$360$	0	0
$361$	−6.59423	−0.347065
$362$	0	0
$363$	−0.388919	−0.0204129
$364$	0	0
$365$	2.70139	0.141397
$366$	0	0
$367$	8.80691	0.459717	0.229858	−	0.973224i	$-0.426174\pi$
$367$	8.80691	0.459717	0.229858	+	0.973224i	$0.426174\pi$
$368$	0	0
$369$	−5.59206	−0.291111
$370$	0	0
$371$	6.80290	0.353189
$372$	0	0
$373$	−15.2503	−0.789631	−0.394815	−	0.918760i	$-0.629191\pi$
$373$	−15.2503	−0.789631	−0.394815	+	0.918760i	$0.629191\pi$
$374$	0	0
$375$	−3.26489	−0.168598
$376$	0	0
$377$	7.34401	0.378236
$378$	0	0
$379$	20.1132	1.03314	0.516572	−	0.856244i	$-0.327208\pi$
$379$	20.1132	1.03314	0.516572	+	0.856244i	$0.327208\pi$
$380$	0	0
$381$	20.2129	1.03554
$382$	0	0
$383$	11.9427	0.610241	0.305121	−	0.952314i	$-0.401303\pi$
$383$	11.9427	0.610241	0.305121	+	0.952314i	$0.401303\pi$
$384$	0	0
$385$	5.11921	0.260899
$386$	0	0
$387$	18.9132	0.961411
$388$	0	0
$389$	−35.4543	−1.79760	−0.898802	−	0.438354i	$-0.855562\pi$
$389$	−35.4543	−1.79760	−0.898802	+	0.438354i	$0.855562\pi$
$390$	0	0
$391$	39.2020	1.98253
$392$	0	0
$393$	14.2370	0.718162
$394$	0	0
$395$	−4.55848	−0.229362
$396$	0	0
$397$	−18.8173	−0.944412	−0.472206	−	0.881488i	$-0.656542\pi$
$397$	−18.8173	−0.944412	−0.472206	+	0.881488i	$0.656542\pi$
$398$	0	0
$399$	19.0027	0.951325
$400$	0	0
$401$	13.8322	0.690745	0.345373	−	0.938466i	$-0.387753\pi$
$401$	13.8322	0.690745	0.345373	+	0.938466i	$0.387753\pi$
$402$	0	0
$403$	−4.25995	−0.212203
$404$	0	0
$405$	0.0502160	0.00249525
$406$	0	0
$407$	−35.2525	−1.74740
$408$	0	0
$409$	9.55163	0.472298	0.236149	−	0.971717i	$-0.424115\pi$
$409$	9.55163	0.472298	0.236149	+	0.971717i	$0.424115\pi$
$410$	0	0
$411$	−16.9436	−0.835768
$412$	0	0
$413$	29.8786	1.47023
$414$	0	0
$415$	−2.61945	−0.128584
$416$	0	0
$417$	−10.0318	−0.491261
$418$	0	0
$419$	−18.3179	−0.894890	−0.447445	−	0.894311i	$-0.647666\pi$
$419$	−18.3179	−0.894890	−0.447445	+	0.894311i	$0.647666\pi$
$420$	0	0
$421$	0.688614	0.0335610	0.0167805	−	0.999859i	$-0.494658\pi$
$421$	0.688614	0.0335610	0.0167805	+	0.999859i	$0.494658\pi$
$422$	0	0
$423$	−9.09549	−0.442238
$424$	0	0
$425$	26.7592	1.29801
$426$	0	0
$427$	41.1360	1.99071
$428$	0	0
$429$	−11.6875	−0.564278
$430$	0	0
$431$	14.0939	0.678880	0.339440	−	0.940628i	$-0.389762\pi$
$431$	14.0939	0.678880	0.339440	+	0.940628i	$0.389762\pi$
$432$	0	0
$433$	−37.0221	−1.77917	−0.889584	−	0.456771i	$-0.849006\pi$
$433$	−37.0221	−1.77917	−0.889584	+	0.456771i	$0.849006\pi$
$434$	0	0
$435$	0.755100	0.0362043
$436$	0	0
$437$	−25.3211	−1.21127
$438$	0	0
$439$	−41.4306	−1.97737	−0.988687	−	0.149992i	$-0.952075\pi$
$439$	−41.4306	−1.97737	−0.988687	+	0.149992i	$0.952075\pi$
$440$	0	0
$441$	−32.6852	−1.55644
$442$	0	0
$443$	−26.4349	−1.25596	−0.627979	−	0.778230i	$-0.716118\pi$
$443$	−26.4349	−1.25596	−0.627979	+	0.778230i	$0.716118\pi$
$444$	0	0
$445$	2.25411	0.106855
$446$	0	0
$447$	−12.3704	−0.585099
$448$	0	0
$449$	5.95194	0.280890	0.140445	−	0.990089i	$-0.455147\pi$
$449$	5.95194	0.280890	0.140445	+	0.990089i	$0.455147\pi$
$450$	0	0
$451$	10.3023	0.485116
$452$	0	0
$453$	−17.8204	−0.837275
$454$	0	0
$455$	4.86821	0.228225
$456$	0	0
$457$	−22.7407	−1.06377	−0.531883	−	0.846818i	$-0.678515\pi$
$457$	−22.7407	−1.06377	−0.531883	+	0.846818i	$0.678515\pi$
$458$	0	0
$459$	−28.4926	−1.32992
$460$	0	0
$461$	22.9724	1.06993	0.534966	−	0.844873i	$-0.320324\pi$
$461$	22.9724	1.06993	0.534966	+	0.844873i	$0.320324\pi$
$462$	0	0
$463$	−8.78401	−0.408228	−0.204114	−	0.978947i	$-0.565431\pi$
$463$	−8.78401	−0.408228	−0.204114	+	0.978947i	$0.565431\pi$
$464$	0	0
$465$	−0.438001	−0.0203118
$466$	0	0
$467$	−20.1370	−0.931829	−0.465914	−	0.884830i	$-0.654275\pi$
$467$	−20.1370	−0.931829	−0.465914	+	0.884830i	$0.654275\pi$
$468$	0	0
$469$	−9.97933	−0.460803
$470$	0	0
$471$	−12.5410	−0.577861
$472$	0	0
$473$	−34.8439	−1.60212
$474$	0	0
$475$	−17.2841	−0.793051
$476$	0	0
$477$	2.49578	0.114274
$478$	0	0
$479$	28.7983	1.31583	0.657914	−	0.753093i	$-0.271439\pi$
$479$	28.7983	1.31583	0.657914	+	0.753093i	$0.271439\pi$
$480$	0	0
$481$	−33.5240	−1.52856
$482$	0	0
$483$	−38.7859	−1.76482
$484$	0	0
$485$	−0.534318	−0.0242621
$486$	0	0
$487$	−3.64463	−0.165154	−0.0825770	−	0.996585i	$-0.526315\pi$
$487$	−3.64463	−0.165154	−0.0825770	+	0.996585i	$0.526315\pi$
$488$	0	0
$489$	11.2068	0.506788
$490$	0	0
$491$	−10.3119	−0.465370	−0.232685	−	0.972552i	$-0.574751\pi$
$491$	−10.3119	−0.465370	−0.232685	+	0.972552i	$0.574751\pi$
$492$	0	0
$493$	−12.4947	−0.562733
$494$	0	0
$495$	1.87808	0.0844135
$496$	0	0
$497$	61.0143	2.73687
$498$	0	0
$499$	−19.6409	−0.879248	−0.439624	−	0.898182i	$-0.644888\pi$
$499$	−19.6409	−0.879248	−0.439624	+	0.898182i	$0.644888\pi$
$500$	0	0
$501$	−14.7346	−0.658294
$502$	0	0
$503$	−16.2133	−0.722914	−0.361457	−	0.932389i	$-0.617720\pi$
$503$	−16.2133	−0.722914	−0.361457	+	0.932389i	$0.617720\pi$
$504$	0	0
$505$	1.19012	0.0529596
$506$	0	0
$507$	2.95059	0.131041
$508$	0	0
$509$	27.4549	1.21692	0.608459	−	0.793585i	$-0.291788\pi$
$509$	27.4549	1.21692	0.608459	+	0.793585i	$0.291788\pi$
$510$	0	0
$511$	44.2257	1.95643
$512$	0	0
$513$	18.4037	0.812545
$514$	0	0
$515$	−5.05796	−0.222880
$516$	0	0
$517$	16.7567	0.736959
$518$	0	0
$519$	−2.77051	−0.121612
$520$	0	0
$521$	−22.1298	−0.969524	−0.484762	−	0.874646i	$-0.661094\pi$
$521$	−22.1298	−0.969524	−0.484762	+	0.874646i	$0.661094\pi$
$522$	0	0
$523$	−3.02989	−0.132488	−0.0662439	−	0.997803i	$-0.521102\pi$
$523$	−3.02989	−0.132488	−0.0662439	+	0.997803i	$0.521102\pi$
$524$	0	0
$525$	−26.4752	−1.15547
$526$	0	0
$527$	7.24764	0.315712
$528$	0	0
$529$	28.6823	1.24706
$530$	0	0
$531$	10.9615	0.475690
$532$	0	0
$533$	9.79715	0.424362
$534$	0	0
$535$	3.35608	0.145096
$536$	0	0
$537$	−14.9701	−0.646005
$538$	0	0
$539$	60.2162	2.59370
$540$	0	0
$541$	15.4448	0.664024	0.332012	−	0.943275i	$-0.392272\pi$
$541$	15.4448	0.664024	0.332012	+	0.943275i	$0.392272\pi$
$542$	0	0
$543$	−9.32145	−0.400022
$544$	0	0
$545$	−1.23392	−0.0528554
$546$	0	0
$547$	−20.5523	−0.878753	−0.439377	−	0.898303i	$-0.644801\pi$
$547$	−20.5523	−0.878753	−0.439377	+	0.898303i	$0.644801\pi$
$548$	0	0
$549$	15.0915	0.644091
$550$	0	0
$551$	8.07050	0.343815
$552$	0	0
$553$	−74.6290	−3.17355
$554$	0	0
$555$	−3.44689	−0.146312
$556$	0	0
$557$	−11.5422	−0.489060	−0.244530	−	0.969642i	$-0.578634\pi$
$557$	−11.5422	−0.489060	−0.244530	+	0.969642i	$0.578634\pi$
$558$	0	0
$559$	−33.1354	−1.40148
$560$	0	0
$561$	19.8845	0.839523
$562$	0	0
$563$	−3.05791	−0.128876	−0.0644378	−	0.997922i	$-0.520525\pi$
$563$	−3.05791	−0.128876	−0.0644378	+	0.997922i	$0.520525\pi$
$564$	0	0
$565$	1.31512	0.0553276
$566$	0	0
$567$	0.822109	0.0345253
$568$	0	0
$569$	21.9008	0.918131	0.459065	−	0.888403i	$-0.348184\pi$
$569$	21.9008	0.918131	0.459065	+	0.888403i	$0.348184\pi$
$570$	0	0
$571$	27.4859	1.15025	0.575124	−	0.818066i	$-0.304954\pi$
$571$	27.4859	1.15025	0.575124	+	0.818066i	$0.304954\pi$
$572$	0	0
$573$	−19.2909	−0.805889
$574$	0	0
$575$	35.2782	1.47120
$576$	0	0
$577$	37.1190	1.54529	0.772643	−	0.634841i	$-0.218934\pi$
$577$	37.1190	1.54529	0.772643	+	0.634841i	$0.218934\pi$
$578$	0	0
$579$	22.9164	0.952373
$580$	0	0
$581$	−42.8842	−1.77914
$582$	0	0
$583$	−4.59798	−0.190429
$584$	0	0
$585$	1.78600	0.0738418
$586$	0	0
$587$	−0.989026	−0.0408215	−0.0204107	−	0.999792i	$-0.506497\pi$
$587$	−0.989026	−0.0408215	−0.0204107	+	0.999792i	$0.506497\pi$
$588$	0	0
$589$	−4.68135	−0.192892
$590$	0	0
$591$	−25.0979	−1.03239
$592$	0	0
$593$	4.85228	0.199259	0.0996296	−	0.995025i	$-0.468234\pi$
$593$	4.85228	0.199259	0.0996296	+	0.995025i	$0.468234\pi$
$594$	0	0
$595$	−8.28250	−0.339549
$596$	0	0
$597$	−12.8192	−0.524654
$598$	0	0
$599$	−7.76350	−0.317208	−0.158604	−	0.987342i	$-0.550699\pi$
$599$	−7.76350	−0.317208	−0.158604	+	0.987342i	$0.550699\pi$
$600$	0	0
$601$	24.9450	1.01753	0.508765	−	0.860906i	$-0.330102\pi$
$601$	24.9450	1.01753	0.508765	+	0.860906i	$0.330102\pi$
$602$	0	0
$603$	−3.66111	−0.149092
$604$	0	0
$605$	−0.109491	−0.00445145
$606$	0	0
$607$	13.1933	0.535498	0.267749	−	0.963489i	$-0.413720\pi$
$607$	13.1933	0.535498	0.267749	+	0.963489i	$0.413720\pi$
$608$	0	0
$609$	12.3621	0.500937
$610$	0	0
$611$	15.9351	0.644665
$612$	0	0
$613$	1.57806	0.0637374	0.0318687	−	0.999492i	$-0.489854\pi$
$613$	1.57806	0.0637374	0.0318687	+	0.999492i	$0.489854\pi$
$614$	0	0
$615$	1.00733	0.0406194
$616$	0	0
$617$	29.0115	1.16796	0.583979	−	0.811769i	$-0.301495\pi$
$617$	29.0115	1.16796	0.583979	+	0.811769i	$0.301495\pi$
$618$	0	0
$619$	37.9483	1.52527	0.762634	−	0.646830i	$-0.223906\pi$
$619$	37.9483	1.52527	0.762634	+	0.646830i	$0.223906\pi$
$620$	0	0
$621$	−37.5634	−1.50737
$622$	0	0
$623$	36.9030	1.47849
$624$	0	0
$625$	23.6170	0.944679
$626$	0	0
$627$	−12.8437	−0.512926
$628$	0	0
$629$	57.0359	2.27417
$630$	0	0
$631$	20.5653	0.818693	0.409347	−	0.912379i	$-0.365757\pi$
$631$	20.5653	0.818693	0.409347	+	0.912379i	$0.365757\pi$
$632$	0	0
$633$	−17.8546	−0.709655
$634$	0	0
$635$	5.69048	0.225820
$636$	0	0
$637$	57.2637	2.26887
$638$	0	0
$639$	22.3843	0.885508
$640$	0	0
$641$	−7.43665	−0.293730	−0.146865	−	0.989157i	$-0.546918\pi$
$641$	−7.43665	−0.293730	−0.146865	+	0.989157i	$0.546918\pi$
$642$	0	0
$643$	−7.87078	−0.310393	−0.155197	−	0.987884i	$-0.549601\pi$
$643$	−7.87078	−0.310393	−0.155197	+	0.987884i	$0.549601\pi$
$644$	0	0
$645$	−3.40693	−0.134148
$646$	0	0
$647$	19.5790	0.769730	0.384865	−	0.922973i	$-0.374248\pi$
$647$	19.5790	0.769730	0.384865	+	0.922973i	$0.374248\pi$
$648$	0	0
$649$	−20.1945	−0.792705
$650$	0	0
$651$	−7.17071	−0.281042
$652$	0	0
$653$	−42.3359	−1.65673	−0.828367	−	0.560186i	$-0.810730\pi$
$653$	−42.3359	−1.65673	−0.828367	+	0.560186i	$0.810730\pi$
$654$	0	0
$655$	4.00811	0.156610
$656$	0	0
$657$	16.2251	0.633000
$658$	0	0
$659$	−29.2661	−1.14005	−0.570024	−	0.821628i	$-0.693066\pi$
$659$	−29.2661	−1.14005	−0.570024	+	0.821628i	$0.693066\pi$
$660$	0	0
$661$	−15.4510	−0.600975	−0.300488	−	0.953786i	$-0.597149\pi$
$661$	−15.4510	−0.600975	−0.300488	+	0.953786i	$0.597149\pi$
$662$	0	0
$663$	18.9095	0.734385
$664$	0	0
$665$	5.34978	0.207456
$666$	0	0
$667$	−16.4725	−0.637818
$668$	0	0
$669$	−4.69907	−0.181677
$670$	0	0
$671$	−27.8033	−1.07333
$672$	0	0
$673$	32.6971	1.26038	0.630190	−	0.776441i	$-0.282977\pi$
$673$	32.6971	1.26038	0.630190	+	0.776441i	$0.282977\pi$
$674$	0	0
$675$	−25.6407	−0.986911
$676$	0	0
$677$	−2.37398	−0.0912396	−0.0456198	−	0.998959i	$-0.514526\pi$
$677$	−2.37398	−0.0912396	−0.0456198	+	0.998959i	$0.514526\pi$
$678$	0	0
$679$	−8.74756	−0.335701
$680$	0	0
$681$	−26.4647	−1.01413
$682$	0	0
$683$	24.0074	0.918618	0.459309	−	0.888277i	$-0.348097\pi$
$683$	24.0074	0.918618	0.459309	+	0.888277i	$0.348097\pi$
$684$	0	0
$685$	−4.77010	−0.182256
$686$	0	0
$687$	20.9017	0.797451
$688$	0	0
$689$	−4.37254	−0.166580
$690$	0	0
$691$	−22.5978	−0.859660	−0.429830	−	0.902910i	$-0.641426\pi$
$691$	−22.5978	−0.859660	−0.429830	+	0.902910i	$0.641426\pi$
$692$	0	0
$693$	30.7469	1.16798
$694$	0	0
$695$	−2.82423	−0.107129
$696$	0	0
$697$	−16.6683	−0.631358
$698$	0	0
$699$	24.4900	0.926296
$700$	0	0
$701$	−11.1286	−0.420322	−0.210161	−	0.977667i	$-0.567399\pi$
$701$	−11.1286	−0.420322	−0.210161	+	0.977667i	$0.567399\pi$
$702$	0	0
$703$	−36.8403	−1.38946
$704$	0	0
$705$	1.63842	0.0617065
$706$	0	0
$707$	19.4840	0.732770
$708$	0	0
$709$	−28.0040	−1.05171	−0.525857	−	0.850573i	$-0.676255\pi$
$709$	−28.0040	−1.05171	−0.525857	+	0.850573i	$0.676255\pi$
$710$	0	0
$711$	−27.3791	−1.02680
$712$	0	0
$713$	9.55499	0.357837
$714$	0	0
$715$	−3.29035	−0.123052
$716$	0	0
$717$	21.9250	0.818804
$718$	0	0
$719$	−10.3463	−0.385851	−0.192925	−	0.981213i	$-0.561798\pi$
$719$	−10.3463	−0.385851	−0.192925	+	0.981213i	$0.561798\pi$
$720$	0	0
$721$	−82.8061	−3.08386
$722$	0	0
$723$	−3.55903	−0.132362
$724$	0	0
$725$	−11.2441	−0.417595
$726$	0	0
$727$	6.46275	0.239690	0.119845	−	0.992793i	$-0.461760\pi$
$727$	6.46275	0.239690	0.119845	+	0.992793i	$0.461760\pi$
$728$	0	0
$729$	17.2611	0.639300
$730$	0	0
$731$	56.3748	2.08510
$732$	0	0
$733$	26.0921	0.963733	0.481866	−	0.876245i	$-0.339959\pi$
$733$	26.0921	0.963733	0.481866	+	0.876245i	$0.339959\pi$
$734$	0	0
$735$	5.88777	0.217174
$736$	0	0
$737$	6.74489	0.248451
$738$	0	0
$739$	25.6074	0.941983	0.470991	−	0.882138i	$-0.343896\pi$
$739$	25.6074	0.941983	0.470991	+	0.882138i	$0.343896\pi$
$740$	0	0
$741$	−12.2139	−0.448689
$742$	0	0
$743$	23.4242	0.859349	0.429675	−	0.902984i	$-0.358628\pi$
$743$	23.4242	0.859349	0.429675	+	0.902984i	$0.358628\pi$
$744$	0	0
$745$	−3.48260	−0.127593
$746$	0	0
$747$	−15.7329	−0.575637
$748$	0	0
$749$	54.9440	2.00761
$750$	0	0
$751$	17.9266	0.654152	0.327076	−	0.944998i	$-0.393937\pi$
$751$	17.9266	0.654152	0.327076	+	0.944998i	$0.393937\pi$
$752$	0	0
$753$	−12.5189	−0.456212
$754$	0	0
$755$	−5.01693	−0.182585
$756$	0	0
$757$	37.4335	1.36055	0.680273	−	0.732959i	$-0.261861\pi$
$757$	37.4335	1.36055	0.680273	+	0.732959i	$0.261861\pi$
$758$	0	0
$759$	26.2149	0.951540
$760$	0	0
$761$	26.1274	0.947119	0.473559	−	0.880762i	$-0.342969\pi$
$761$	26.1274	0.947119	0.473559	+	0.880762i	$0.342969\pi$
$762$	0	0
$763$	−20.2011	−0.731328
$764$	0	0
$765$	−3.03859	−0.109861
$766$	0	0
$767$	−19.2044	−0.693429
$768$	0	0
$769$	−30.6997	−1.10706	−0.553530	−	0.832830i	$-0.686719\pi$
$769$	−30.6997	−1.10706	−0.553530	+	0.832830i	$0.686719\pi$
$770$	0	0
$771$	0.0721218	0.00259740
$772$	0	0
$773$	35.2397	1.26748	0.633742	−	0.773545i	$-0.281518\pi$
$773$	35.2397	1.26748	0.633742	+	0.773545i	$0.281518\pi$
$774$	0	0
$775$	6.52221	0.234285
$776$	0	0
$777$	−56.4306	−2.02443
$778$	0	0
$779$	10.7663	0.385743
$780$	0	0
$781$	−41.2387	−1.47564
$782$	0	0
$783$	11.9724	0.427860
$784$	0	0
$785$	−3.53065	−0.126014
$786$	0	0
$787$	28.2856	1.00827	0.504136	−	0.863624i	$-0.331811\pi$
$787$	28.2856	1.00827	0.504136	+	0.863624i	$0.331811\pi$
$788$	0	0
$789$	−25.1079	−0.893866
$790$	0	0
$791$	21.5305	0.765535
$792$	0	0
$793$	−26.4400	−0.938913
$794$	0	0
$795$	−0.449578	−0.0159449
$796$	0	0
$797$	47.7702	1.69211	0.846054	−	0.533098i	$-0.178972\pi$
$797$	47.7702	1.69211	0.846054	+	0.533098i	$0.178972\pi$
$798$	0	0
$799$	−27.1111	−0.959121
$800$	0	0
$801$	13.5386	0.478363
$802$	0	0
$803$	−29.8915	−1.05485
$804$	0	0
$805$	−10.9193	−0.384855
$806$	0	0
$807$	−13.9181	−0.489942
$808$	0	0
$809$	0.539249	0.0189590	0.00947950	−	0.999955i	$-0.496983\pi$
$809$	0.539249	0.0189590	0.00947950	+	0.999955i	$0.496983\pi$
$810$	0	0
$811$	15.5609	0.546419	0.273209	−	0.961955i	$-0.411915\pi$
$811$	15.5609	0.546419	0.273209	+	0.961955i	$0.411915\pi$
$812$	0	0
$813$	−27.1978	−0.953870
$814$	0	0
$815$	3.15501	0.110515
$816$	0	0
$817$	−36.4133	−1.27394
$818$	0	0
$819$	29.2393	1.02171
$820$	0	0
$821$	−31.1606	−1.08751	−0.543757	−	0.839243i	$-0.682999\pi$
$821$	−31.1606	−1.08751	−0.543757	+	0.839243i	$0.682999\pi$
$822$	0	0
$823$	6.47505	0.225706	0.112853	−	0.993612i	$-0.464001\pi$
$823$	6.47505	0.225706	0.112853	+	0.993612i	$0.464001\pi$
$824$	0	0
$825$	17.8942	0.622997
$826$	0	0
$827$	−38.4566	−1.33727	−0.668633	−	0.743593i	$-0.733120\pi$
$827$	−38.4566	−1.33727	−0.668633	+	0.743593i	$0.733120\pi$
$828$	0	0
$829$	44.8537	1.55783	0.778917	−	0.627127i	$-0.215769\pi$
$829$	44.8537	1.55783	0.778917	+	0.627127i	$0.215769\pi$
$830$	0	0
$831$	−6.96869	−0.241741
$832$	0	0
$833$	−97.4253	−3.37559
$834$	0	0
$835$	−4.14820	−0.143554
$836$	0	0
$837$	−6.94470	−0.240044
$838$	0	0
$839$	−29.9583	−1.03427	−0.517137	−	0.855902i	$-0.673002\pi$
$839$	−29.9583	−1.03427	−0.517137	+	0.855902i	$0.673002\pi$
$840$	0	0
$841$	−23.7498	−0.818958
$842$	0	0
$843$	27.2366	0.938079
$844$	0	0
$845$	0.830673	0.0285760
$846$	0	0
$847$	−1.79253	−0.0615921
$848$	0	0
$849$	8.11612	0.278545
$850$	0	0
$851$	75.1938	2.57761
$852$	0	0
$853$	8.98166	0.307526	0.153763	−	0.988108i	$-0.450861\pi$
$853$	8.98166	0.307526	0.153763	+	0.988108i	$0.450861\pi$
$854$	0	0
$855$	1.96267	0.0671219
$856$	0	0
$857$	−53.4838	−1.82697	−0.913486	−	0.406869i	$-0.866620\pi$
$857$	−53.4838	−1.82697	−0.913486	+	0.406869i	$0.866620\pi$
$858$	0	0
$859$	−45.4016	−1.54908	−0.774541	−	0.632523i	$-0.782019\pi$
$859$	−45.4016	−1.54908	−0.774541	+	0.632523i	$0.782019\pi$
$860$	0	0
$861$	16.4914	0.562026
$862$	0	0
$863$	34.9575	1.18997	0.594984	−	0.803738i	$-0.297159\pi$
$863$	34.9575	1.18997	0.594984	+	0.803738i	$0.297159\pi$
$864$	0	0
$865$	−0.779975	−0.0265199
$866$	0	0
$867$	−13.7788	−0.467954
$868$	0	0
$869$	50.4407	1.71108
$870$	0	0
$871$	6.41418	0.217336
$872$	0	0
$873$	−3.20921	−0.108615
$874$	0	0
$875$	−15.0479	−0.508712
$876$	0	0
$877$	−13.7854	−0.465501	−0.232750	−	0.972537i	$-0.574773\pi$
$877$	−13.7854	−0.465501	−0.232750	+	0.972537i	$0.574773\pi$
$878$	0	0
$879$	33.1017	1.11649
$880$	0	0
$881$	−43.8060	−1.47586	−0.737930	−	0.674877i	$-0.764197\pi$
$881$	−43.8060	−1.47586	−0.737930	+	0.674877i	$0.764197\pi$
$882$	0	0
$883$	54.1930	1.82374	0.911869	−	0.410481i	$-0.134639\pi$
$883$	54.1930	1.82374	0.911869	+	0.410481i	$0.134639\pi$
$884$	0	0
$885$	−1.97456	−0.0663742
$886$	0	0
$887$	−21.2735	−0.714293	−0.357146	−	0.934048i	$-0.616250\pi$
$887$	−21.2735	−0.714293	−0.357146	+	0.934048i	$0.616250\pi$
$888$	0	0
$889$	93.1613	3.12453
$890$	0	0
$891$	−0.555652	−0.0186150
$892$	0	0
$893$	17.5114	0.585997
$894$	0	0
$895$	−4.21448	−0.140874
$896$	0	0
$897$	24.9295	0.832373
$898$	0	0
$899$	−3.04542	−0.101571
$900$	0	0
$901$	7.43920	0.247836
$902$	0	0
$903$	−55.7765	−1.85612
$904$	0	0
$905$	−2.62424	−0.0872328
$906$	0	0
$907$	24.9100	0.827123	0.413561	−	0.910476i	$-0.364285\pi$
$907$	24.9100	0.827123	0.413561	+	0.910476i	$0.364285\pi$
$908$	0	0
$909$	7.14807	0.237087
$910$	0	0
$911$	23.6340	0.783029	0.391515	−	0.920172i	$-0.371951\pi$
$911$	23.6340	0.783029	0.391515	+	0.920172i	$0.371951\pi$
$912$	0	0
$913$	28.9849	0.959259
$914$	0	0
$915$	−2.71852	−0.0898716
$916$	0	0
$917$	65.6185	2.16691
$918$	0	0
$919$	19.0938	0.629845	0.314922	−	0.949117i	$-0.398022\pi$
$919$	19.0938	0.629845	0.314922	+	0.949117i	$0.398022\pi$
$920$	0	0
$921$	−14.7145	−0.484858
$922$	0	0
$923$	−39.2167	−1.29083
$924$	0	0
$925$	51.3271	1.68763
$926$	0	0
$927$	−30.3790	−0.997778
$928$	0	0
$929$	−0.695777	−0.0228277	−0.0114138	−	0.999935i	$-0.503633\pi$
$929$	−0.695777	−0.0228277	−0.0114138	+	0.999935i	$0.503633\pi$
$930$	0	0
$931$	62.9283	2.06239
$932$	0	0
$933$	−14.5938	−0.477778
$934$	0	0
$935$	5.59803	0.183075
$936$	0	0
$937$	49.6710	1.62268	0.811340	−	0.584574i	$-0.198738\pi$
$937$	49.6710	1.62268	0.811340	+	0.584574i	$0.198738\pi$
$938$	0	0
$939$	−29.4124	−0.959838
$940$	0	0
$941$	−48.7061	−1.58777	−0.793886	−	0.608066i	$-0.791946\pi$
$941$	−48.7061	−1.58777	−0.793886	+	0.608066i	$0.791946\pi$
$942$	0	0
$943$	−21.9748	−0.715600
$944$	0	0
$945$	7.93630	0.258168
$946$	0	0
$947$	−45.9269	−1.49242	−0.746211	−	0.665709i	$-0.768129\pi$
$947$	−45.9269	−1.49242	−0.746211	+	0.665709i	$0.768129\pi$
$948$	0	0
$949$	−28.4259	−0.922744
$950$	0	0
$951$	−8.28476	−0.268652
$952$	0	0
$953$	35.4799	1.14931	0.574653	−	0.818397i	$-0.305137\pi$
$953$	35.4799	1.14931	0.574653	+	0.818397i	$0.305137\pi$
$954$	0	0
$955$	−5.43092	−0.175740
$956$	0	0
$957$	−8.35536	−0.270090
$958$	0	0
$959$	−78.0934	−2.52177
$960$	0	0
$961$	−29.2335	−0.943016
$962$	0	0
$963$	20.1573	0.649559
$964$	0	0
$965$	6.45160	0.207684
$966$	0	0
$967$	32.6857	1.05110	0.525551	−	0.850762i	$-0.323859\pi$
$967$	32.6857	1.05110	0.525551	+	0.850762i	$0.323859\pi$
$968$	0	0
$969$	20.7801	0.667552
$970$	0	0
$971$	−16.1543	−0.518417	−0.259208	−	0.965821i	$-0.583462\pi$
$971$	−16.1543	−0.518417	−0.259208	+	0.965821i	$0.583462\pi$
$972$	0	0
$973$	−46.2368	−1.48228
$974$	0	0
$975$	17.0168	0.544975
$976$	0	0
$977$	38.8081	1.24158	0.620790	−	0.783977i	$-0.286812\pi$
$977$	38.8081	1.24158	0.620790	+	0.783977i	$0.286812\pi$
$978$	0	0
$979$	−24.9423	−0.797158
$980$	0	0
$981$	−7.41115	−0.236620
$982$	0	0
$983$	−9.68349	−0.308855	−0.154428	−	0.988004i	$-0.549353\pi$
$983$	−9.68349	−0.308855	−0.154428	+	0.988004i	$0.549353\pi$
$984$	0	0
$985$	−7.06574	−0.225133
$986$	0	0
$987$	26.8233	0.853796
$988$	0	0
$989$	74.3222	2.36331
$990$	0	0
$991$	−50.5002	−1.60419	−0.802097	−	0.597194i	$-0.796282\pi$
$991$	−50.5002	−1.60419	−0.802097	+	0.597194i	$0.796282\pi$
$992$	0	0
$993$	−5.07468	−0.161040
$994$	0	0
$995$	−3.60895	−0.114411
$996$	0	0
$997$	2.79261	0.0884430	0.0442215	−	0.999022i	$-0.485919\pi$
$997$	2.79261	0.0884430	0.0442215	+	0.999022i	$0.485919\pi$
$998$	0	0
$999$	−54.6519	−1.72911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.19	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.19	✓	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)

\(f(q)\)	\(=\)	\(q-1.08193 q^{3} -0.304592 q^{5} -4.98661 q^{7} -1.82943 q^{9} +O(q^{10})\)	\(q-1.08193 q^{3} -0.304592 q^{5} -4.98661 q^{7} -1.82943 q^{9} +3.37038 q^{11} +3.20513 q^{13} +0.329546 q^{15} -5.45302 q^{17} +3.52218 q^{19} +5.39515 q^{21} -7.18904 q^{23} -4.90722 q^{25} +5.22509 q^{27} +2.29133 q^{29} -1.32910 q^{31} -3.64650 q^{33} +1.51888 q^{35} -10.4595 q^{37} -3.46771 q^{39} +3.05671 q^{41} -10.3383 q^{43} +0.557231 q^{45} +4.97175 q^{47} +17.8663 q^{49} +5.89977 q^{51} -1.36423 q^{53} -1.02659 q^{55} -3.81074 q^{57} -5.99176 q^{59} -8.24929 q^{61} +9.12268 q^{63} -0.976256 q^{65} +2.00122 q^{67} +7.77802 q^{69} -12.2356 q^{71} -8.86889 q^{73} +5.30926 q^{75} -16.8068 q^{77} +14.9659 q^{79} -0.164863 q^{81} +8.59988 q^{83} +1.66095 q^{85} -2.47905 q^{87} -7.40042 q^{89} -15.9827 q^{91} +1.43799 q^{93} -1.07283 q^{95} +1.75421 q^{97} -6.16589 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

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